Rosetta Code

Non-continuous subsequences

Consider some sequence of elements. (It differs from a mere set of elements by having an ordering among members.)

Task
Non-continuous subsequences
You are encouraged to solve this task according to the task description, using any language you may know.

A subsequence contains some subset of the elements of this sequence, in the same order.

A continuous subsequence is one in which no elements are missing between the first and last elements of the subsequence.

Note: Subsequences are defined structurally, not by their contents. So a sequence a,b,c,d will always have the same subsequences and continuous subsequences, no matter which values are substituted; it may even be the same value.


Task: Find all non-continuous subsequences for a given sequence.


Example

For the sequence   1,2,3,4,   there are five non-continuous subsequences, namely:

  •   1,3
  •   1,4
  •   2,4
  •   1,3,4
  •   1,2,4


Goal

There are different ways to calculate those subsequences.

Demonstrate algorithm(s) that are natural for the language.

Other tasks related to string operations:
Metrics
Counting
Remove/replace
Anagrams/Derangements/shuffling
Find/Search/Determine
Formatting
Song lyrics/poems/Mad Libs/phrases
Tokenize
Sequences



11l

Translation of: Python
F ncsub(seq, s = 0)
   I seq.empty
      R I s >= 3 {[[Int]()]} E [[Int]]()
   E
      V x = seq[0.<1]
      V xs = seq[1..]
      V p2 = s % 2
      V p1 = !p2
      R ncsub(xs, s + p1).map(ys -> @x + ys) [+] ncsub(xs, s + p2)

print(ncsub(Array(1..3)))
print(ncsub(Array(1..4)))
print(ncsub(Array(1..5)))
Output:
[[1, 3]]
[[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]]
[[1, 2, 3, 5], [1, 2, 4, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4, 5], [1, 3, 4], [1, 3, 5], [1, 3], [1, 4, 5], [1, 4], [1, 5], [2, 3, 5], [2, 4, 5], [2, 4], [2, 5], [3, 5]]

Ada

Recursive

with Ada.Text_IO;  use Ada.Text_IO;

procedure Test_Non_Continuous is
   type Sequence is array (Positive range <>) of Integer;
   procedure Put_NCS
             (  Tail : Sequence;                -- To generate subsequences of
                Head : Sequence := (1..0 => 1); -- Already generated
                Contiguous : Boolean := True    -- It is still continuous
             )  is
   begin
      if not Contiguous and then Head'Length > 1 then
         for I in Head'Range loop
            Put (Integer'Image (Head (I)));
         end loop;
         New_Line;
      end if;
      if Tail'Length /= 0 then 
         declare
            New_Head : Sequence (Head'First..Head'Last + 1);
         begin
            New_Head (Head'Range) := Head;
            for I in Tail'Range loop
               New_Head (New_Head'Last) := Tail (I);
               Put_NCS
               (  Tail => Tail (I + 1..Tail'Last),
                  Head => New_Head,
                  Contiguous => Contiguous and then (I = Tail'First or else Head'Length = 0)
               );
            end loop;
         end;
      end if;
   end Put_NCS;
begin
   Put_NCS ((1,2,3));     New_Line;
   Put_NCS ((1,2,3,4));   New_Line;
   Put_NCS ((1,2,3,4,5)); New_Line;
end Test_Non_Continuous;
Output:
 1 3

 1 2 4
 1 3
 1 3 4
 1 4
 2 4

 1 2 3 5
 1 2 4
 1 2 4 5
 1 2 5
 1 3
 1 3 4
 1 3 4 5
 1 3 5
 1 4
 1 4 5
 1 5
 2 3 5
 2 4
 2 4 5
 2 5
 3 5

ALGOL 68

Recursive

Translation of: Ada

- note: This specimen retains the original Ada coding style.

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d
PROC test non continuous = VOID: BEGIN
   MODE SEQMODE = CHAR;
   MODE SEQ = [1:0]SEQMODE;
   MODE YIELDSEQ = PROC(SEQ)VOID;

   PROC gen ncs =
             (  SEQ tail,       # To generate subsequences of #
                SEQ head,       #           Already generated #
                BOOL contiguous,#      It is still continuous #
                YIELDSEQ yield
             )  VOID:
   BEGIN
      IF NOT contiguous ANDTH UPB head > 1 THEN
         yield (head)
      FI;
      IF UPB tail /= 0 THEN 
            [UPB head+1]SEQMODE new head;
            new head [:UPB head] := head;
            FOR i TO UPB tail DO
               new head [UPB new head] := tail [i];
               gen ncs
               (  tail[i + 1:UPB tail],
                  new head,
                  contiguous ANDTH (i = LWB tail OREL UPB head = 0),
                  yield
               )
            OD
      FI
   END # put ncs #;

 # FOR SEQ seq IN # gen ncs(("a","e","i","o","u"), (), TRUE, # ) DO ( #
 ##   (SEQ seq)VOID:
      print((seq, new line))
 # OD # )
END; test non continuous
Output:
aeiu
aeo
aeou
aeu
ai
aio
aiou
aiu
ao
aou
au
eiu
eo
eou
eu
iu

Iterative

Translation of: C

- note: This specimen retains the original C coding style.

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d

Note: This specimen can only handle sequences of length less than bits width of bits. 

MODE SEQMODE = STRING;
MODE SEQ = [1:0]SEQMODE;
MODE YIELDSEQ = PROC(SEQ)VOID;

PROC gen ncs = (SEQ seq, YIELDSEQ yield)VOID:
BEGIN
  IF UPB seq - 1 > bits width THEN stop FI;
  [UPB seq]SEQMODE out;  INT upb out;

  BITS lim := 16r1 SHL UPB seq;
  BITS upb k := lim SHR 1;
  # assert(lim); #

  BITS empty = 16r000000000; # const #

  FOR j TO ABS lim-1 DO
    INT state := 1;
    BITS k1 := upb k; 
    WHILE k1 NE empty DO
      BITS b := BIN j AND k1;
      CASE state IN
        # state 1 # IF b NE empty THEN state +:= 1 FI,
        # state 2 # IF b EQ empty THEN state +:= 1 FI,
        # state 3 # 
          BEGIN
            IF b EQ empty THEN GO TO continue k1 FI;
            upb out := 0; 
            BITS k2 := upb k; FOR i WHILE k2 NE empty DO
              IF (BIN j AND k2) NE empty THEN out[upb out +:= 1] := seq[i] FI;
              k2 := k2 SHR 1
            OD;
            yield(out[:upb out]);
            k1 := empty # empty: ending containing loop #
         END
      ESAC;
      continue k1: k1 := k1 SHR 1
    OD
  OD
END;

main:(
  []STRING seqs = ("a","e","i","o","u");
# FOR SEQ seq IN # gen ncs(seqs, # ) DO ( #
##   (SEQ seq)VOID:
    print((seq, new line))
# OD # )
)
Output:
iu
eu
eo
eou
eiu
au
ao
aou
ai
aiu
aio
aiou
aeu
aeo
aeou
aeiu

AutoHotkey

using filtered templates ahk forum: discussion 

MsgBox % noncontinuous("a,b,c,d,e", ",")
MsgBox % noncontinuous("1,2,3,4", ",")

noncontinuous(list, delimiter)
{
stringsplit, seq, list, %delimiter%
n := seq0                                            ; sequence length
Loop % x := (1<<n) - 1 {                                  ; try all 0-1 candidate sequences
   If !RegExMatch(b:=ToBin(A_Index,n),"^0*1*0*$") {  ; drop continuous subsequences
      Loop Parse, b
         t .= A_LoopField ? seq%A_Index% " " : ""         ; position -> number
		 t .= "`n"                                   ; new sequences in new lines
   }
}
return t
}

ToBin(n,W=16) {  ; LS W-bits of Binary representation of n
   Return W=1 ? n&1 : ToBin(n>>1,W-1) . n&1
}

BBC BASIC

Works with: BBC BASIC for Windows
      DIM list1$(3)
      list1$() = "1", "2", "3", "4"
      PRINT "For [1, 2, 3, 4] non-continuous subsequences are:"
      PROCnon_continuous_subsequences(list1$())
      DIM list2$(4)
      list2$() = "1", "2", "3", "4", "5"
      PRINT "For [1, 2, 3, 4, 5] non-continuous subsequences are:"
      PROCnon_continuous_subsequences(list2$())
      END
      
      DEF PROCnon_continuous_subsequences(l$())
      LOCAL i%, j%, g%, n%, r%, s%, w%, a$, b$
      n% = DIM(l$(),1)
      FOR s% = 0 TO n%-2
        FOR g% = s%+1 TO n%-1
          a$ = "["
          FOR i% = s% TO g%-1
            a$ += l$(i%) + ", "
          NEXT
          FOR w% = 1 TO n%-g%
            r% = n%+1-g%-w%
            FOR i% = 1 TO 2^r%-1 STEP 2
              b$ = a$
              FOR j% = 0 TO r%-1
                IF i% AND 2^j% b$ += l$(g%+w%+j%) + ", "
              NEXT
              PRINT LEFT$(LEFT$(b$)) + "]"
            NEXT i%
          NEXT w%
        NEXT g%
      NEXT s%
      ENDPROC
Output:
For [1, 2, 3, 4] non-continuous subsequences are:
[1, 3]
[1, 3, 4]
[1, 4]
[1, 2, 4]
[2, 4]
For [1, 2, 3, 4, 5] non-continuous subsequences are:
[1, 3]
[1, 3, 4]
[1, 3, 5]
[1, 3, 4, 5]
[1, 4]
[1, 4, 5]
[1, 5]
[1, 2, 4]
[1, 2, 4, 5]
[1, 2, 5]
[1, 2, 3, 5]
[2, 4]
[2, 4, 5]
[2, 5]
[2, 3, 5]
[3, 5]

Bracmat 

( ( noncontinuous
  =   sub
    .     ( sub
          =   su a nc
            .   !arg:(?su.?nc)
              &   !su
                :   %
                    %?a
                    ( %:[%(sub$(!sjt.!nc !a))
                    |   ?
                      & !nc:~
                      & out$(!nc !a)
                      & ~
                    )
          )
        & sub$(dummy !arg.)
      | 
  )
& noncontinuous$(e r n i t)
);
Output:
e n t
e n
e n i
e n i t
e i
e i t
e t
e r i
e r i t
e r t
e r n t
r i
r i t
r t
r n t
n t

C

Note: This specimen can only handle lists of length less than the number of bits in an int. 

#include <assert.h>
#include <stdio.h>

int main(int c, char **v)
{
	unsigned int n = 1 << (c - 1), i = n, j, k;
	assert(n);

	while (i--) {
		if (!(i & (i + (i & -(int)i)))) // consecutive 1s
			continue;

		for (j = n, k = 1; j >>= 1; k++)
			if (i & j) printf("%s ", v[k]);

		putchar('\n');
	}

	return 0;
}

Example use: 

$ ./noncont 1 2 3 4
1 2 4 
1 3 4 
1 3 
2 4 
1 4 
$ ./noncont 1 2 3 4 5 6 7 8 9 0 | wc -l
968

Using "consecutive + gap + any subsequence" to produce disjointed sequences: 

#include <assert.h>
#include <stdio.h>
#include <stdlib.h>

void binprint(unsigned int n, unsigned int m)
{
	char c[sizeof(n) * 8 + 1];
	int i = 0;
	while (m >>= 1)	c[i++] = n & m ? '#' : '-';
	c[i] = 0;
	puts(c);
}

int main(int c, char **v)
{
	unsigned int n, gap, left, right;
	if (c < 2 || ! (n = 1 << atoi(v[1]))) n = 16;

	for (gap = 2; gap < n; gap <<= 1)
		for (left = gap << 1; left < n; left |= left << 1)
			for (right = 1; right < gap; right++)
				binprint(left | right, n);

	return 0;
}

Recursive method

Using recursion and a state transition table. 

#include <stdio.h>

typedef unsigned char sint;
enum states { s_blnk = 0, s_tran, s_cont, s_disj };

/* Recursively look at each item in list, taking both choices of
   picking the item or not.  The state at each step depends on prvious
   pickings, with the state transition table:
	blank + no pick -> blank
	blank + pick -> contiguous
	transitional + no pick -> transitional
	transitional + pick -> disjoint
	contiguous + no pick -> transitional
	contiguous + pick -> contiguous
	disjoint + pick -> disjoint
	disjoint + no pick -> disjoint
   At first step, before looking at any item, state is blank.
   Because state is known at each step and needs not be calculated,
   it can be quite fast.
*/
unsigned char tbl[][2] = {
	{ s_blnk, s_cont },
	{ s_tran, s_disj },
	{ s_tran, s_cont },
	{ s_disj, s_disj },
};

void pick(sint n, sint step, sint state, char **v, unsigned long bits)
{
	int i, b;
	if (step == n) {
		if (state != s_disj) return;
		for (i = 0, b = 1; i < n; i++, b <<= 1)
			if ((b & bits)) printf("%s ", v[i]);
		putchar('\n');
		return;
	}

	bits <<= 1;
	pick(n, step + 1, tbl[state][0], v, bits); /* no pick */
	pick(n, step + 1, tbl[state][1], v, bits | 1); /* pick */
}

int main(int c, char **v)
{
	if (c - 1 >= sizeof(unsigned long) * 4)
		printf("Too many items");
	else
		pick(c - 1, 0, s_blnk, v + 1, 0);
	return 0;
}

running it: 

% ./a.out 1 2 3 4
1 3 
1 4 
2 4 
1 2 4 
1 3 4
% ./a.out 1 2 3 4 5 6 7 8 9 0 | wc -l
968

C# 

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    public static void Main() {
        var sequence = new[] { "A", "B", "C", "D" };
        foreach (var subset in Subsets(sequence.Length).Where(s => !IsContinuous(s))) {
            Console.WriteLine(string.Join(" ", subset.Select(i => sequence[i])));
        }
    }
    
    static IEnumerable<List<int>> Subsets(int length) {
        int[] values = Enumerable.Range(0, length).ToArray();
        var stack = new Stack<int>(length);
        for (int i = 0; stack.Count > 0 || i < length; ) {
            if (i < length) {
                stack.Push(i++);
                yield return (from index in stack.Reverse() select values[index]).ToList();
            } else {
                i = stack.Pop() + 1;
                if (stack.Count > 0) i = stack.Pop() + 1;
            }
        }
    }

    static bool IsContinuous(List<int> list) => list[list.Count - 1] - list[0] + 1 == list.Count;

}
Output:
A B D
A C
A C D
A D
B D

C++ 

/*
 * Nigel Galloway, July 19th., 2017 - Yes well is this any better?
 */
class N{
  uint n,i,g,e,l;
public:
  N(uint n): n(n-1),i{},g{},e(1),l(n-1){}
  bool hasNext(){
    g=(1<<n)+e;for(i=l;i<n;++i) g+=1<<i;
    if (l==2)             {l=--n; e=1; return true;}
    if (e<((1<<(l-1))-1)) {++e;        return true;}
                           e=1; --l;   return (l>0);
  }
  uint next() {return g;}
};

Which may be used as follows: 

int main(){
  N n(4);
  while (n.hasNext()) std::cout << n.next() << "\t* " << std::bitset<4>(n.next()) << std::endl;
}
Output:
9       * 1001
10      * 1010
11      * 1011
13      * 1101
5       * 0101

I can count the length of the sequence: 

int main(){
  N n(31);
  int z{};for (;n.hasNext();++z); std::cout << z << std::endl;
}
Output:
2147483151

Clojure

Here's a simple approach that uses the clojure.contrib.combinatorics library to generate subsequences, and then filters out the continuous subsequences using a naïve subseq test: 

(use '[clojure.contrib.combinatorics :only (subsets)])

(defn of-min-length [min-length]
  (fn [s] (>= (count s) min-length)))

(defn runs [c l]
  (map (partial take l) (take-while not-empty (iterate rest c))))

(defn is-subseq? [c sub]
  (some identity (map = (runs c (count sub)) (repeat sub))))

(defn non-continuous-subsequences [s]
  (filter (complement (partial is-subseq? s)) (subsets s)))


(filter (of-min-length 2) (non-continuous-subsequences [:a :b :c :d]))

CoffeeScript

Use binary bitmasks to enumerate our sequences. 

is_contigous_binary = (n) ->
  # return true if binary representation of n is
  # of the form 1+0+
  # examples:
  #     0 true
  #     1 true
  #   100 true
  #   110 true
  #  1001 false
  #  1010 false

  # special case zero, or you'll get an infinite loop later
  return true if n == 0

  # first remove 0s from end
  while n % 2 == 0
    n = n / 2
  
  # next, take advantage of the fact that a continuous
  # run of 1s would be of the form 2^n - 1
  is_power_of_two(n + 1)

is_power_of_two = (m) ->
  while m % 2 == 0
    m = m / 2
  m == 1

seq_from_bitmap = (arr, n) ->
  # grabs elements from array according to a bitmap
  # e.g. if n == 13 (1101), and arr = ['a', 'b', 'c', 'd'],
  # then return ['a', 'c', 'd'] (flipping bits to 1011, so
  # that least significant bit comes first)
  i = 0
  new_arr = []
  while n > 0
    if n % 2 == 1
      new_arr.push arr[i]
      n -= 1
    n /= 2
    i += 1
  new_arr

non_contig_subsequences = (arr) ->
  # Return all subsqeuences from an array that have a "hole" in
  # them.  The order of the subsequences is not specified here.
    
  # This algorithm uses binary counting, so it is limited to
  # small lists, but large lists would be unwieldy regardless.
  bitmasks = [0...Math.pow(2, arr.length)]
  (seq_from_bitmap arr, n for n in bitmasks when !is_contigous_binary n)

arr = [1,2,3,4]
console.log non_contig_subsequences arr
for n in [1..10]
  arr = [1..n]
  num_solutions = non_contig_subsequences(arr).length
  console.log "for n=#{n} there are #{num_solutions} solutions"
Output:
> coffee non_contig_subseq.coffee 
[ [ 1, 3 ],
  [ 1, 4 ],
  [ 2, 4 ],
  [ 1, 2, 4 ],
  [ 1, 3, 4 ] ]
for n=1 there are 0 solutions
for n=2 there are 0 solutions
for n=3 there are 1 solutions
for n=4 there are 5 solutions
for n=5 there are 16 solutions
for n=6 there are 42 solutions
for n=7 there are 99 solutions
for n=8 there are 219 solutions
for n=9 there are 466 solutions
for n=10 there are 968 solutions

Common Lisp 

(defun all-subsequences (list)
  (labels ((subsequences (tail &optional (acc '()) (result '()))
             "Return a list of the subsequence designators of the
              subsequences of tail. Each subsequence designator is a
              list of tails of tail, the subsequence being the first
              element of each tail."
             (if (endp tail)
               (list* (reverse acc) result)
               (subsequences (rest tail) (list* tail acc)
                             (append (subsequences (rest tail) acc) result))))
           (continuous-p (subsequence-d)
             "True if the designated subsequence is continuous."
             (loop for i in subsequence-d
                   for j on (first subsequence-d)
                   always (eq i j)))
           (designated-sequence (subsequence-d)
             "Destructively transforms a subsequence designator into
              the designated subsequence."
             (map-into subsequence-d 'first subsequence-d)))
    (let ((nc-subsequences (delete-if #'continuous-p (subsequences list))))
      (map-into nc-subsequences #'designated-sequence nc-subsequences))))
Translation of: Scheme
(defun all-subsequences2 (list)
  (labels ((recurse (s list)
             (if (endp list)
                 (if (>= s 3)
                     '(())
                     '())
                 (let ((x (car list))
                       (xs (cdr list)))
                   (if (evenp s)
                       (append (mapcar (lambda (ys) (cons x ys))
                                       (recurse (+ s 1) xs))
                               (recurse s xs))
                       (append (mapcar (lambda (ys) (cons x ys))
                                       (recurse s xs))
                               (recurse (+ s 1) xs)))))))
    (recurse 0 list)))

D

Recursive Version

Translation of: Python
T[][] ncsub(T)(in T[] seq, in uint s=0) pure nothrow @safe {
    if (seq.length) {
        typeof(return) aux;
        foreach (ys; ncsub(seq[1 .. $], s + !(s % 2)))
            aux ~= seq[0] ~ ys;
        return aux ~ ncsub(seq[1 .. $], s + s % 2);
    } else
        return new typeof(return)(s >= 3, 0);
}

void main() @safe {
    import std.stdio;

    [1, 2, 3].ncsub.writeln;
    [1, 2, 3, 4].ncsub.writeln;
    foreach (const nc; [1, 2, 3, 4, 5].ncsub)
        nc.writeln;
}
Output:
[[1, 3]]
[[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]]
[1, 2, 3, 5]
[1, 2, 4, 5]
[1, 2, 4]
[1, 2, 5]
[1, 3, 4, 5]
[1, 3, 4]
[1, 3, 5]
[1, 3]
[1, 4, 5]
[1, 4]
[1, 5]
[2, 3, 5]
[2, 4, 5]
[2, 4]
[2, 5]
[3, 5]

Faster Lazy Version

This version doesn't copy the sub-arrays. 

struct Ncsub(T) {
    T[] seq;

    int opApply(int delegate(ref T[]) dg) const {
        immutable n = seq.length;
        int result;
        auto S = new T[n];

        OUTER: foreach (immutable i; 1 .. 1 << n) {
            uint lenS;
            bool nc = false;
            foreach (immutable j; 0 .. n + 1) {
                immutable k = i >> j;
                if (k == 0) {
                    if (nc) {
                        auto auxS = S[0 .. lenS];
                        result = dg(auxS);
                        if (result)
                            break OUTER;
                    }
                    break;
                } else if (k % 2) {
                    S[lenS] = seq[j];
                    lenS++;
                } else if (lenS)
                    nc = true;
            }
        }

        return result;
    }
}

void main() {
    import std.array, std.range;

    //assert(24.iota.array.Ncsub!int.walkLength == 16_776_915);
    auto r = 24.iota.array;
    uint counter = 0;
    foreach (s; Ncsub!int(r))
        counter++;
    assert(counter == 16_776_915);
}

Generator Version

This version doesn't copy the sub-arrays, and it's a little slower than the opApply-based version. 

import std.stdio, std.array, std.range, std.concurrency;

Generator!(T[]) ncsub(T)(in T[] seq) {
    return new typeof(return)({
        immutable n = seq.length;
        auto S = new T[n];

        foreach (immutable i; 1 .. 1 << n) {
            uint lenS = 0;
            bool nc = false;
            foreach (immutable j; 0 .. n + 1) {
                immutable k = i >> j;
                if (k == 0) {
                    if (nc)
                        yield(S[0 .. lenS]);
                    break;
                } else if (k % 2) {
                    S[lenS] = seq[j];
                    lenS++;
                } else if (lenS)
                    nc = true;
            }
        }
    });
}

void main() {
    assert(24.iota.array.ncsub.walkLength == 16_776_915);

    [1, 2, 3].ncsub.writeln;
    [1, 2, 3, 4].ncsub.writeln;
    foreach (const nc; [1, 2, 3, 4, 5].ncsub)
        nc.writeln;
}

EasyLang

Translation of: Nim
func[][] ncsub seq[] s .
   if len seq[] = 0
      if s >= 3
         return [ [ ] ]
      .
      return [ ]
   .
   last = seq[$]
   len seq[] -1
   res[][] = ncsub seq[] (s + s mod 2)
   r[][] = ncsub seq[] (s + 1 - s mod 2)
   for i to len r[][]
      r[i][] &= last
      res[][] &= r[i][]
   .
   return res[][]
.
print ncsub [ 1 2 3 4 ] 0
Output:
[
 [ 1 3 ]
 [ 1 4 ]
 [ 2 4 ]
 [ 1 2 4 ]
 [ 1 3 4 ]
]

Elixir

Translation of: Erlang
defmodule RC do
  defp masks(n) do
    maxmask = trunc(:math.pow(2, n)) - 1
    Enum.map(3..maxmask, &Integer.to_string(&1, 2))
    |> Enum.filter_map(&contains_noncont(&1), &String.rjust(&1, n, ?0)) # padding
  end
 
  defp contains_noncont(n) do
    Regex.match?(~r/10+1/, n)
  end
 
  defp apply_mask_to_list(mask, list) do
    Enum.zip(to_char_list(mask), list)
    |> Enum.filter_map(fn {include, _} -> include > ?0 end, fn {_, value} -> value end)
  end

  def ncs(list) do
    Enum.map(masks(length(list)), fn mask -> apply_mask_to_list(mask, list) end)
  end
end

IO.inspect RC.ncs([1,2,3])
IO.inspect RC.ncs([1,2,3,4])
IO.inspect RC.ncs('abcd')
Output:
[[1, 3]]
[[2, 4], [1, 4], [1, 3], [1, 3, 4], [1, 2, 4]]
['bd', 'ad', 'ac', 'acd', 'abd']

Erlang

Erlang's not optimized for strings or math, so this is pretty inefficient. Nonetheless, it works by generating the set of all possible "bitmasks" (represented as strings), filters for those with non-continuous subsequences, and maps from that set over the list. One immediate point for optimization that would complicate the code a bit would be to compile the regular expression, the problem being where you'd put it. 

-module(rosetta).
-export([ncs/1]).

masks(N) ->
    MaxMask = trunc(math:pow(2, N)),
    Total = lists:map(fun(X) -> integer_to_list(X, 2) end,
                lists:seq(3, MaxMask)),
    Filtered = lists:filter(fun(X) -> contains_noncont(X) end, Total),
    lists:map(fun(X) -> string:right(X, N, $0) end, Filtered). % padding
   
contains_noncont(N) ->
    case re:run(N, "10+1") of
        {match, _} -> true;
        nomatch -> false
    end.

apply_mask_to_list(Mask, List) ->
    Zipped = lists:zip(Mask, List),
    Filtered = lists:filter(fun({Include, _}) -> Include > 48 end, Zipped),
    lists:map(fun({_, Value}) -> Value end, Filtered).

ncs(List) ->
    lists:map(fun(Mask) -> apply_mask_to_list(Mask, List) end,
                masks(length(List))).
Output:
Eshell V5.10.1  (abort with ^G)
1> c(rosetta).
{ok,rosetta}
2> rosetta:ncs([1,2,3,4]).
[[2,4],[1,4],[1,3],[1,3,4],[1,2,4]]

F#

Generate only the non-continuous subsequences

(*
  A function to generate only the non-continuous subsequences.
  Nigel Galloway July 20th., 2017
*)
let N n =
  let     fn n = Seq.map (fun g->(2<<<n)+g)
  let rec fg n = seq{if n>0 then yield! seq{1..((1<<<n)-1)}|>fn n; yield! fg (n-1)|>fn n}
  Seq.collect fg ({1..(n-2)})

This may be used as follows: 

let Ng ng = N ng |> Seq.iter(fun n->printf "%2d -> " n; {0..(ng-1)}|>Seq.iter (fun g->if (n&&&(1<<<g))>0 then printf "%d " (g+1));printfn "")
Ng 4
Output:
 5 -> 1 3 
 9 -> 1 4 
10 -> 2 4 
11 -> 1 2 4 
13 -> 1 3 4

Counting the number of non-continuous subsequences is interesting: 

> Seq.length (N 20);;                                                                 
Real: 00:00:00.169, CPU: 00:00:00.169, GC gen0: 0, gen1: 0
val it : int = 1048365
> Seq.length (N 23);;
Real: 00:00:01.238, CPU: 00:00:01.239, GC gen0: 0, gen1: 0
val it : int = 8388331
> Seq.length (N 24);;
Real: 00:00:02.520, CPU: 00:00:02.523, GC gen0: 0, gen1: 0
val it : int = 16776915
> Seq.length (N 25);;
Real: 00:00:04.926, CPU: 00:00:04.930, GC gen0: 0, gen1: 0
val it : int = 33554106

Generate all subsequences and filter out the continuous

(*
  A function to filter out continuous subsequences.
  Nigel Galloway July 24th., 2017
*)
let Nonseq n=
  let fn = function
    |((n,0),true )->(n+1,1)
    |((n,_),false)->(n,0)
    |(n,_)        ->n
  {5..(1<<<n)-1}|>Seq.choose(fun i->if fst({0..n-1}|>Seq.takeWhile(fun n->(1<<<(n-1))<i)|>Seq.fold(fun n g->fn (n,(i&&&(1<<<g)>0)))(0,0)) > 1 then Some(i) else None)

Again counting the number of non-continuous subsequences 

> Seq.length (Nonseq 20);;
Real: 00:00:02.356, CPU: 00:00:02.389, GC gen0: 183, gen1: 0
val it : int = 1048365
> Seq.length (Nonseq 23);;
Real: 00:00:20.714, CPU: 00:00:20.950, GC gen0: 1571, gen1: 0
val it : int = 8388331
> Seq.length (Nonseq 24);;
Real: 00:00:43.129, CPU: 00:00:43.601, GC gen0: 3216, gen1: 0
val it : int = 16776915
> Seq.length (Nonseq 25);;
Real: 00:01:28.853, CPU: 00:01:29.869, GC gen0: 6577, gen1: 0
val it : int = 33554106

Conclusion

Find a better filter or use the generator.


FreeBASIC

Translation of: BBC BASIC
Sub Subsecuencias_no_continuas(l() As String)
    Dim As Integer i, j, g, n, r, s, w
    Dim As String a, b, c
    n = Ubound(l)
    For s = 0 To n-2
        For g = s+1 To n-1
            a = "["
            For i = s To g-1
                a += l(i) + ", "
            Next i
            For w = 1 To n-g
                r = n+1-g-w
                For i = 1 To 2^r-1 Step 2
                    b = a
                    For j = 0 To r-1
                        If i And 2^j Then b += l(g+w+j) + ", "
                    Next j
                    'Print Left(Left(b)) + "]"
                    c = (Left(b, Len (b)-1))
                    Print Left(c, Len(c)-1) + "]"
                Next i
            Next w
        Next g
    Next s
End Sub

Dim lista1(3) As String = {"1", "2", "3", "4"}
Print "Para [1, 2, 3, 4] las subsecuencias no continuas son:"
Subsecuencias_no_continuas(lista1())
Dim lista2(4) As String = {"e", "r", "n", "i", "t"}
Print "Para [e, r, n, i, t] las subsecuencias no continuas son:"
Subsecuencias_no_continuas(lista2())
Sleep
Output:
Para [1, 2, 3, 4] las subsecuencias no continuas son:
[1, 3]
[1, 3, 4]
[1, 4]
[1, 2, 4]
[2, 4]
Para [e, r, n, i, t] las subsecuencias no continuas son:
[e, n]
[e, n, i]
[e, n, t]
[e, n, i, t]
[e, i]
[e, i, t]
[e, t]
[e, r, i]
[e, r, i, t]
[e, r, t]
[e, r, n, t]
[r, i]
[r, i, t]
[r, t]
[r, n, t]
[n, t]


Go

Generate the power set (power sequence, actually) with a recursive function, but keep track of the state of the subsequence on the way down. When you get to the bottom, if state == non-continuous, then include the subsequence. It's just filtering merged in with generation. 

package main

import "fmt"

const ( // state:
    m   = iota // missing:  all elements missing so far
    c          // continuous:  all elements included so far are continuous
    cm         // one or more continuous followed by one or more missing
    cmc        // non-continuous subsequence
)

func ncs(s []int) [][]int {
    if len(s) < 3 {
        return nil
    }
    return append(n2(nil, s[1:], m), n2([]int{s[0]}, s[1:], c)...)
}

var skip = []int{m, cm, cm, cmc}
var incl = []int{c, c, cmc, cmc}

func n2(ss, tail []int, seq int) [][]int {
    if len(tail) == 0 {
        if seq != cmc {
            return nil
        }
        return [][]int{ss}
    }
    return append(n2(append([]int{}, ss...), tail[1:], skip[seq]),
        n2(append(ss, tail[0]), tail[1:], incl[seq])...)
}

func main() {
    ss := ncs([]int{1, 2, 3, 4})
    fmt.Println(len(ss), "non-continuous subsequences:")
    for _, s := range ss {
        fmt.Println("  ", s)
    }
}
Output:
5 non-continuous subsequences:
   [2 4]
   [1 4]
   [1 3]
   [1 3 4]
   [1 2 4]

Haskell

Generalized monadic filter

action p x = if p x then succ x else x

fenceM p q s []     = guard (q s) >> return []
fenceM p q s (x:xs) = do
  (f,g) <- p 
  ys <- fenceM p q (g s) xs
  return $ f x ys

ncsubseq = fenceM [((:), action even), (flip const, action odd)] (>= 3) 0
Output:
*Main> ncsubseq [1..3]
[[1,3]]
*Main> ncsubseq [1..4]
[[1,2,4],[1,3,4],[1,3],[1,4],[2,4]]
*Main> ncsubseq [1..5]
[[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3],[1,4,5],[1,4],[1,5],[2,3,5],[2,4,5],[2,4],[2,5],[3,5]]

Filtered templates

This implementation works by computing templates of all possible subsequences of the given length of sequence, discarding the continuous ones, then applying the remaining templates to the input list. 

continuous = null . dropWhile not . dropWhile id . dropWhile not
ncs xs = map (map fst . filter snd . zip xs) $
           filter (not . continuous) $
             mapM (const [True,False]) xs

Recursive

Recursive method with powerset as helper function. 

import Data.List

poset = foldr (\x p -> p ++ map (x:) p) [[]]

ncsubs [] = [[]]
ncsubs (x:xs) = tail $ nc [x] xs
  where
    nc [_] [] = [[]]
    nc (_:x:xs) [] = nc [x] xs
    nc  xs (y:ys) = (nc (xs++[y]) ys) ++ map (xs++) (tail $ poset ys)
Output:
 *Main> ncsubs "aaa"
 ["aa"]
 (0.00 secs, 0 bytes)
 *Main> ncsubs [9..12]
 [[10,12],[9,10,12],[9,12],[9,11],[9,11,12]]
 (0.00 secs, 522544 bytes)
 *Main> ncsubs []
 [[]]
 (0.00 secs, 0 bytes)
 *Main> ncsubs [1]
 []
 (0.00 secs, 0 bytes)

A disjointed subsequence is a consecutive subsequence followed by a gap, then by any nonempty subsequence to its right: 

import Data.List (subsequences, tails, delete)

disjoint a = concatMap (cutAt a) [1..length a - 2] where
	cutAt s n = [a ++ b |	b <- delete [] (subsequences right),
				a <- init (tails left) ] where
		(left, _:right) = splitAt n s

main = print $ length $ disjoint [1..20]

Build a lexicographic list of consecutive subsequences, and a list of all subsequences, then subtract one from the other: 

import Data.List (inits, tails)

subseqs = foldr (\x s -> [x] : map (x:) s ++ s) []

consecs = concatMap (tail.inits) . tails

minus [] [] = []
minus (a:as) bb@(b:bs)
	| a == b = minus as bs
	| otherwise = a:minus as bb

disjoint s = (subseqs s) `minus` (consecs s)

main = mapM_ print $ disjoint [1..4]

J

We select those combinations where the end of the first continuous subsequence appears before the start of the last continuous subsequence: 

allmasks=: 2 #:@i.@^ #
firstend=:1 0 i.&1@E."1 ]
laststart=: 0 1 {:@I.@E."1 ]
noncont=: <@#~ (#~ firstend < laststart)@allmasks

Example use: 

   noncont 1+i.4
┌───┬───┬───┬─────┬─────┐
│2 4│1 4│1 3│1 3 4│1 2 4│
└───┴───┴───┴─────┴─────┘
   noncont 'aeiou'
┌──┬──┬──┬───┬───┬──┬──┬───┬──┬───┬───┬────┬───┬───┬────┬────┐
│iu│eu│eo│eou│eiu│au│ao│aou│ai│aiu│aio│aiou│aeu│aeo│aeou│aeiu│
└──┴──┴──┴───┴───┴──┴──┴───┴──┴───┴───┴────┴───┴───┴────┴────┘
   #noncont i.10
968

Alternatively, since there are relatively few continuous sequences, we could specifically exclude them: 

contmasks=: a: ;@,  1 <:/~@i.&.>@i.@+ #
noncont=: <@#~ (allmasks -. contmasks)

(we get the same behavior from this implementation)

Java 

public class NonContinuousSubsequences {

    public static void main(String args[]) {
        seqR("1234", "", 0, 0);
    }

    private static void seqR(String s, String c, int i, int added) {
        if (i == s.length()) {
            if (c.trim().length() > added)
                System.out.println(c);
        } else {
            seqR(s, c + s.charAt(i), i + 1, added + 1);
            seqR(s, c + ' ', i + 1, added);
        }
    }
}

12 4
1 34
1 3 
1  4
 2 4

JavaScript

Uses powerset() function from here. Uses a JSON stringifier from http://www.json.org/js.html

Works with: SpiderMonkey
function non_continuous_subsequences(ary) {
    var non_continuous = new Array();
    for (var i = 0; i < ary.length; i++) {
        if (! is_array_continuous(ary[i])) {
            non_continuous.push(ary[i]);
        }
    }
    return non_continuous;
}

function is_array_continuous(ary) {
    if (ary.length < 2)
        return true;
    for (var j = 1; j < ary.length; j++) {
        if (ary[j] - ary[j-1] != 1) {
            return false;
        }
    }
    return true;
}

load('json2.js'); /* http://www.json.org/js.html */

print(JSON.stringify( non_continuous_subsequences( powerset([1,2,3,4]))));
Output:
[[1,3],[1,4],[2,4],[1,2,4],[1,3,4]]

jq

Works with: jq version 1.4

In order to handle arrays of more than a handful of elements, we define non_continuous_subsequences/0 as a generator; that is, it produces a stream of arrays, each of which is a non-continuous subsequence of the given sequence.

Since the non-continuous subsequences are dense in the set of all subsets, we will use the powerset approach, and accordingly begin by defining subsets/0 as a generator. 

# Generate a stream of subsets of the input array
def subsets:
  if length == 0 then []
  else .[0] as $first
    | (.[1:] | subsets) 
    | ., ([$first] + .)
  end ;

# Generate a stream of non-continuous indices in the range 0 <= i < .
def non_continuous_indices:
  [range(0;.)] | subsets
  | select(length > 1 and length != 1 + .[length-1] - .[0]) ;

def non_continuous_subsequences:
  (length | non_continuous_indices) as $ix
  | [.[ $ix[] ]] ;

Example: To show that the above approach can be used for relatively large n, let us count the number of non-continuous subsequences of [0, 1, ..., 19]. 

def count(f): reduce f as $i (0; . + 1);

count( [range(0;20)] | non_continuous_subsequences)
Output:
$ jq -n -f powerset_generator.jq
1048365

Julia

This solution uses an iterator over non-contiguous sub-sequences, NCSubSeq. In the spirit of Julia's permutations and combinations built-ins, NCSubSeq provides an array of indices that can be used to create each subsequence from the full sequence. Sub-sequences are indexed by integers whose bit patterns indicate which members are included.

NCSubSeq works by filtering indices according to whether all 1s in these indices have bit pattern that are contiguous (using the iscontseq functions). This is an easy to implement approach. Greater efficiency might be achieved by exploiting the property that a sequence is contiguous if and only if its index is a difference of two powers of 2. This property is used to create the length(NCSubSeq(n)) function, which gives the number of non-contiguous sub-sequences of a sequence of length n.

NCSubSeq works transparently for sequence lengths up to WORD_SIZE-1 (typically 63). It can be extended to work for longer sequences by casting n to a larger integer, e.g. using Big(n). A more polished implementation would handle this extension behind the scenes.

Iterator and Functions 

using Printf, IterTools

import Base.IteratorSize, Base.iterate, Base.length 

iscontseq(n::Integer) = count_zeros(n) == leading_zeros(n) + trailing_zeros(n)
iscontseq(n::BigInt)  = !ismatch(r"0", rstrip(bin(n), '0'))
 
function makeint2seq(n::Integer)
    idex = collect(1:n)
    function int2seq(m::Integer)
        d = digits(m, base=2, pad=n)
        idex[d .== 1]
    end
    return int2seq
end
 
struct NCSubSeq{T<:Integer}
    n::T
end
 
mutable struct NCSubState{T<:Integer}
    m::T
    m2s::Function
end
 
Base.IteratorSize(::NCSubSeq) = Base.HasLength()
Base.length(a::NCSubSeq) = 2 ^ a.n - a.n * (a.n + 1) ÷ 2 - 1
function Base.iterate(a::NCSubSeq, as::NCSubState=NCSubState(5, makeint2seq(a.n)))
    if 2 ^ a.n - 3 < as.m
        return nothing
    end
    s = as.m2s(as.m)
    as.m += 1
    while iscontseq(as.m)
        as.m += 1
    end
    return (s, as)
end
 
n = 4
println("Testing NCSubSeq for ", n, " items:\n ", join(NCSubSeq(n), " "))
 
s = "Rosetta"
cs = split(s, "")
m = 10
n = length(NCSubSeq(length(cs))) - m
println("\nThe first and last ", m, " NC sub-sequences of \"", s, "\":")
for (i, a) in enumerate(NCSubSeq(length(s)))
    i <= m || n < i || continue
    println(@sprintf "%6d %s" i join(cs[a], ""))
    i == m || continue
    println("    .. ......")
end

println("\nThe first and last ", m, " NC sub-sequences of \"", s, "\"")
for x in IterTools.Iterators.flatten([1:10; 20:10:40; big.(50:50:200)])
    @printf "%7d → %d\n" x length(NCSubSeq(x))
end
Output:
Testing NCSubSeq for 4 items:
 [1, 3] [1, 4] [2, 4] [1, 2, 4] [1, 3, 4]

The first and last 10 NC sub-sequences of "Rosetta":
     1 Rs
     2 Re
     3 oe
     4 Roe
     5 Rse
     6 Rt
     7 ot
     8 Rot
     9 st
    10 Rst
    .. ......
    90 otta
    91 Rotta
    92 stta
    93 Rstta
    94 ostta
    95 Rostta
    96 Retta
    97 oetta
    98 Roetta
    99 Rsetta

The first and last 10 NC sub-sequences of "Rosetta"
      1 → 0
      2 → 0
      3 → 1
      4 → 5
      5 → 16
      6 → 42
      7 → 99
      8 → 219
      9 → 466
     10 → 968
     20 → 1048365
     30 → 1073741358
     40 → 1099511626955
     50 → 1125899906841348
    100 → 1267650600228229401496703200325
    150 → 1427247692705959881058285969449495136382735298
    200 → 1606938044258990275541962092341162602522202993782792835281275

Kotlin 

// version 1.1.2

fun <T> ncs(a: Array<T>) { 
    fun generate(m: Int, k: Int, c: IntArray) {
        if (k == m) {
            if (c[m - 1] != c[0] + m - 1) {
                for (i in 0 until m)  print("${a[c[i]]} ")                
                println()
            }
        }
        else { 
            for (j in 0 until a.size) {
                if (k == 0 || j > c[k - 1]) {
                    c[k] = j
                    generate(m, k + 1, c)
                }
            }
        }
    }

    for (m in 2 until a.size) {
        val c = IntArray(m) 
        generate(m, 0, c)
    }    
}  

fun main(args: Array<String>) {
    val a = arrayOf(1, 2, 3, 4)
    ncs(a)
    println()
    val ca = arrayOf('a', 'b', 'c', 'd', 'e')
    ncs(ca)
}
Output:
1 3 
1 4 
2 4 
1 2 4 
1 3 4 

a c 
a d 
a e 
b d 
b e 
c e 
a b d 
a b e 
a c d 
a c e 
a d e 
b c e 
b d e 
a b c e 
a b d e 
a c d e

M2000 Interpreter 

Module Non_continuous_subsequences (item$(), display){
	Function positions(n) {
		function onebit {
			=lambda b=false (&c)-> {
				=b :if c then  b~:c=not b
			}
		}
		dim k(n)=onebit(), p(n)
		m=true
		flush
		for i=1 to 2^n {
			for j=0 to n-1 :p(j)= k(j)(&m) :next 
			m1=p(0)
			m2=0
			for j=1 to n-1
				if m2 then if m1>p(j) then m2=2:exit for
				if m1 < p(j) then m2++
				m1=p(j)
			next
			if m2=2 then data cons(p())' push a copy of p() to end of stack
			m=true
		}
		=array([])
	}

	a=positions(len(item$()))
	if display then
		For i=0 to len(a)-1
			b=array(a,i)
			line$=format$("{0::-5})",i+1,)
			for j=0 to len(b)-1
				if array(b,j) then line$+=" "+item$(j)
			next
			print line$
			doc$<=line$+{
			}
		next
	end if
	line$="Non continuous subsequences:"+str$(len(a))
	Print line$
	doc$<=line$+{
	}	
}
global doc$
document doc$   ' change string to document object
Non_continuous_subsequences ("1","2","3","4"), true
Non_continuous_subsequences ("a","e","i","o","u"), true
Non_continuous_subsequences ("R","o","s","e","t","t","a"), true
Non_continuous_subsequences ("1","2","3","4","5","6","7","8","9","0"), false
clipboard doc$
Output:
    1) 1 3
    2) 1 4
    3) 2 4
    4) 1 2 4
    5) 1 3 4
Non continuous subsequences: 5
    1) a i
    2) a o
    3) e o
    4) a e o
    5) a i o
    6) a u
    7) e u
    8) a e u
    9) i u
   10) a i u
   11) e i u
   12) a e i u
   13) a o u
   14) e o u
   15) a e o u
   16) a i o u
Non continuous subsequences: 16
    1) R s
    2) R e
    3) o e
    4) R o e
    5) R s e
    6) R t
    7) o t
    8) R o t
    9) s t
   10) R s t
   11) o s t
   12) R o s t
   13) R e t
   14) o e t
   15) R o e t
   16) R s e t
   17) R t
   18) o t
   19) R o t
   20) s t
   21) R s t
   22) o s t
   23) R o s t
   24) e t
   25) R e t
   26) o e t
   27) R o e t
   28) s e t
   29) R s e t
   30) o s e t
   31) R o s e t
   32) R t t
   33) o t t
   34) R o t t
   35) s t t
   36) R s t t
   37) o s t t
   38) R o s t t
   39) R e t t
   40) o e t t
   41) R o e t t
   42) R s e t t
   43) R a
   44) o a
   45) R o a
   46) s a
   47) R s a
   48) o s a
   49) R o s a
   50) e a
   51) R e a
   52) o e a
   53) R o e a
   54) s e a
   55) R s e a
   56) o s e a
   57) R o s e a
   58) t a
   59) R t a
   60) o t a
   61) R o t a
   62) s t a
   63) R s t a
   64) o s t a
   65) R o s t a
   66) e t a
   67) R e t a
   68) o e t a
   69) R o e t a
   70) s e t a
   71) R s e t a
   72) o s e t a
   73) R o s e t a
   74) R t a
   75) o t a
   76) R o t a
   77) s t a
   78) R s t a
   79) o s t a
   80) R o s t a
   81) e t a
   82) R e t a
   83) o e t a
   84) R o e t a
   85) s e t a
   86) R s e t a
   87) o s e t a
   88) R o s e t a
   89) R t t a
   90) o t t a
   91) R o t t a
   92) s t t a
   93) R s t t a
   94) o s t t a
   95) R o s t t a
   96) R e t t a
   97) o e t t a
   98) R o e t t a
   99) R s e t t a
Non continuous subsequences: 99
Non continuous subsequences: 968

Mathematica /Wolfram Language

We make all the subsets then filter out the continuous ones: 

GoodBad[i_List]:=Not[MatchQ[Differences[i],{1..}|{}]]
n=5
Select[Subsets[Range[n]],GoodBad]
Output:
{{1,3},{1,4},{1,5},{2,4},{2,5},{3,5},{1,2,4},{1,2,5},{1,3,4},{1,3,5},{1,4,5},{2,3,5},{2,4,5},{1,2,3,5},{1,2,4,5},{1,3,4,5}}

Nim

Translation of: Python
import sequtils

proc ncsub[T](se: seq[T], s = 0): seq[seq[T]] =
  result = @[]
  if se.len > 0:
    let
      x = se[0..0]
      xs = se[1 .. ^1]
      p2 = s mod 2
      p1 = (s + 1) mod 2
    for ys in ncsub(xs, s + p1):
      result.add(x & ys)
    result.add(ncsub(xs, s + p2))
  elif s >= 3:
    result.add(@[])

echo "ncsub(", toSeq 1.. 3, ") = ", ncsub(toSeq 1..3)
echo "ncsub(", toSeq 1.. 4, ") = ", ncsub(toSeq 1..4)
echo "ncsub(", toSeq 1.. 5, ") = ", ncsub(toSeq 1..5)
Output:
ncsub(@[1, 2, 3]) = @[@[1, 3]]
ncsub(@[1, 2, 3, 4]) = @[@[1, 2, 4], @[1, 3, 4], @[1, 3], @[1, 4], @[2, 4]]
ncsub(@[1, 2, 3, 4, 5]) = @[@[1, 2, 3, 5], @[1, 2, 4, 5], @[1, 2, 4], @[1, 2, 5], @[1, 3, 4, 5], @[1, 3, 4], @[1, 3, 5], @[1, 3], @[1, 4, 5], @[1, 4], @[1, 5], @[2, 3, 5], @[2, 4, 5], @[2, 4], @[2, 5], @[3, 5]]

OCaml

Translation of: Generalized monadic filter
let rec fence s = function
    [] ->
      if s >= 3 then
        [[]]
      else
        []

  | x :: xs ->
      if s mod 2 = 0 then
        List.map
          (fun ys -> x :: ys)
          (fence (s + 1) xs)
        @
          fence s xs
      else
        List.map
          (fun ys -> x :: ys)
          (fence s xs)
        @
          fence (s + 1) xs

let ncsubseq = fence 0
Output:
# ncsubseq [1;2;3];;
- : int list list = [[1; 3]]
# ncsubseq [1;2;3;4];;
- : int list list = [[1; 2; 4]; [1; 3; 4]; [1; 3]; [1; 4]; [2; 4]]
# ncsubseq [1;2;3;4;5];;
- : int list list =
[[1; 2; 3; 5]; [1; 2; 4; 5]; [1; 2; 4]; [1; 2; 5]; [1; 3; 4; 5]; [1; 3; 4];
 [1; 3; 5]; [1; 3]; [1; 4; 5]; [1; 4]; [1; 5]; [2; 3; 5]; [2; 4; 5]; 
 [2; 4]; [2; 5]; [3; 5]]

Oz

A nice application of finite set constraints. We just describe what we want and the constraint system will deliver it: 

declare
  fun {NCSubseq SeqList}
     Seq = {FS.value.make SeqList}
     proc {Script Result}
        %% the result is a subset of Seq
        {FS.subset Result Seq}

        %% at least one element of Seq is missing
        local Gap in
           {FS.include Gap Seq}
           {FS.exclude Gap Result}
           %% and this element is between the smallest
           %% and the largest elements of the subsequence
           Gap >: {FS.int.min Result}
           Gap <: {FS.int.max Result}
        end
      
        %% enumerate all such sets
        {FS.distribute naive [Result]}
     end
  in
     {Map {SearchAll Script} FS.reflect.lowerBoundList}
  end
in
  {Inspect {NCSubseq [1 2 3 4]}}

PARI/GP

Just a simple script, but it's I/O bound so efficiency isn't a concern. (Almost all subsequences are non-contiguous so looping over all possibilities isn't that bad. For length 20 about 99.98% of subsequences are non-contiguous.) 

noncontig(n)=n>>=valuation(n,2);n++;n>>=valuation(n,2);n>1;
nonContigSubseq(v)={
  for(i=5,2^#v-1,
    if(noncontig(i),
      print(vecextract(v,i))
    )
  )
};
nonContigSubseq([1,2,3])
nonContigSubseq(["a","b","c","d","e"])
Output:
[1, 3]

["a", "c"]
["a", "d"]
["b", "d"]
["a", "b", "d"]
["a", "c", "d"]
["a", "e"]
["b", "e"]
["a", "b", "e"]
["c", "e"]
["a", "c", "e"]
["b", "c", "e"]
["a", "b", "c", "e"]
["a", "d", "e"]
["b", "d", "e"]
["a", "b", "d", "e"]
["a", "c", "d", "e"]

Perl 

my ($max, @current);
sub non_continuous {
        my ($idx, $has_gap) = @_;
        my $found;

        for ($idx .. $max) {
                push @current, $_;
                # print "@current\n" if $has_gap; # uncomment for huge output
                $found ++ if $has_gap;
                $found += non_continuous($_ + 1, $has_gap)   if $_ < $max;
                pop @current;
                $has_gap = @current;   # don't set gap flag if it's empty still
        }
        $found;
}

$max = 20;
print "found ", non_continuous(1), " sequences\n";
Output:
found 1048365 sequences

Phix

Straightforward recursive implementation, the only minor trick is that a gap does not mean non-contiguous until you actually take something later.
Counts non-contiguous subsequences of sequences of length 1..20 in under half a second 

with javascript_semantics
integer count = 0
 
procedure ncs(sequence rest, object taken, integer ri=0, bool contig=false, bool gap=false)
    if ri>=length(rest) then
        if contig then
            if integer(taken) then
                count += 1
            else
                ?taken
            end if
        end if
    else
        ri += 1
        ncs(rest,iff(integer(taken)?taken+1:deep_copy(taken)&rest[ri]),ri,gap,gap)
        ncs(rest,taken,ri,contig,iff(integer(taken)?taken!=0:length(taken)!=0))
    end if
end procedure
 
ncs({1,2,3},{})
?"==="
ncs({1,2,3,4},{})
?"==="
atom t0 = time()
sequence s = {}
for i=1 to 20 do
    count = 0
    ncs(tagset(i),0)
    s = append(s,count)
end for
?elapsed(time()-t0)
pp(s)
Output:
{1,3}
"==="
{1,2,4}
{1,3,4}
{1,3}
{1,4}
{2,4}
"==="
"0.3s"
{0,0,1,5,16,42,99,219,466,968,1981,4017,8100,16278,32647,65399,130918,261972,524097,1048365}

Picat

This approach uses power_set/1 (from the util module) to get the proper indices. 

import util.

go =>
  println(1..4=non_cont(1..4)),
  L = "abcde".reverse(),
  println(L=non_cont(L)),
  println(ernit=non_cont("ernit")),
  println(aaa=non_cont("aaa")),
  println(aeiou=non_cont("aeiou")),
  nl,

  println("Printing just the lengths for 1..N for N = 1..20:"), 
  foreach(N in 1..20) 
    println(1..N=non_cont(1..N).length) % just the length
  end,
  nl.

% get all the non-continuous subsequences
non_cont(L) = [ [L[I] : I in S] : S in non_cont_ixs(L.length)].

% get all the index positions that are non-continuous
non_cont_ixs(N) = [ P:  P in power_set(1..N), length(P) > 1, P.last() - P.first() != P.length-1].
Output:
[1,2,3,4] = [[2,4],[1,4],[1,3],[1,3,4],[1,2,4]]
edcba = [ca,da,db,dba,dca,ea,eb,eba,ec,eca,ecb,ecba,eda,edb,edba,edca]
ernit = [nt,rt,ri,rit,rnt,et,ei,eit,en,ent,eni,enit,ert,eri,erit,ernt]
aaa = [aa]
aeiou = [iu,eu,eo,eou,eiu,au,ao,aou,ai,aiu,aio,aiou,aeu,aeo,aeou,aeiu]

Printing just the lengths for 1..N for N = 1..20:
[1] = 0
[1,2] = 0
[1,2,3] = 1
[1,2,3,4] = 5
[1,2,3,4,5] = 16
[1,2,3,4,5,6] = 42
[1,2,3,4,5,6,7] = 99
[1,2,3,4,5,6,7,8] = 219
[1,2,3,4,5,6,7,8,9] = 466
[1,2,3,4,5,6,7,8,9,10] = 968
[1,2,3,4,5,6,7,8,9,10,11] = 1981
[1,2,3,4,5,6,7,8,9,10,11,12] = 4017
[1,2,3,4,5,6,7,8,9,10,11,12,13] = 8100
[1,2,3,4,5,6,7,8,9,10,11,12,13,14] = 16278
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] = 32647
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] = 65399
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] = 130918
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] = 261972
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] = 524097
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] = 1048365

PicoLisp

Translation of: Scheme
(de ncsubseq (Lst)
   (let S 0
      (recur (S Lst)
         (ifn Lst
            (and (>= S 3) '(NIL))
            (let (X (car Lst)  XS (cdr Lst))
               (ifn (bit? 1 S)  # even
                  (conc
                     (mapcar '((YS) (cons X YS))
                        (recurse (inc S) XS) )
                     (recurse S XS) )
                  (conc
                     (mapcar '((YS) (cons X YS))
                        (recurse S XS) )
                     (recurse (inc S) XS) ) ) ) ) ) ) )

Pop11

We modify classical recursive generation of subsets, using variables to keep track if subsequence is continuous. 

define ncsubseq(l);
    lvars acc = [], gap_started = false, is_continuous = true;
    define do_it(l1, l2);
        dlocal gap_started;
        lvars el, save_is_continuous = is_continuous;
        if l2 = [] then
            if not(is_continuous) then
                cons(l1, acc) -> acc;
            endif;
        else
            front(l2) -> el;
            back(l2) -> l2;
            not(gap_started) and is_continuous -> is_continuous;
            do_it(cons(el, l1), l2);
            save_is_continuous -> is_continuous;
            not(l1 = []) or gap_started -> gap_started;
            do_it(l1, l2);
        endif;
    enddefine;
    do_it([], rev(l));
    acc;
enddefine;

ncsubseq([1 2 3 4 5]) =>
Output:
[[1 3] [1 4] [2 4] [1 2 4] [1 3 4] [1 5] [2 5] [1 2 5] [3 5] [1 3 5]
         [2 3 5] [1 2 3 5] [1 4 5] [2 4 5] [1 2 4 5] [1 3 4 5]]

PowerShell 

Function SubSequence ( [Array] $S, [Boolean] $all=$false )
{
   $sc = $S.count
   if( $sc -gt ( 2 - [Int32] $all ) ) {
      [void] $sc--
      0..$sc | ForEach-Object {
         $gap = $_
         "$( $S[ $_ ] )"
         if( $gap -lt $sc )
         {
            SubSequence ( ( $gap + 1 )..$sc | Where-Object { $_ -ne $gap } ) ( ( $gap -ne 0 ) -or $all ) | ForEach-Object {
               [String]::Join( ',', ( ( [String]$_ ).Split(',') | ForEach-Object {
                  $lt = $true
               } {
                  if( $lt -and ( $_ -gt $gap ) )
                  {
                     $S[ $gap ]
                     $lt = $false
                  }
                  $S[ $_ ]
               } {
                  if( $lt )
                  {
                     $S[ $gap ]
                  }
               }
               ) )
            }
         }
      }
      #[String]::Join( ',', $S)
   } else { 
      $S | ForEach-Object { [String] $_ } 
   }
}

Function NonContinuous-SubSequence ( [Array] $S )
{
   $sc = $S.count
   if( $sc -eq 3 )
   {
      [String]::Join( ',', $S[ ( 0,2 ) ] )
   } elseif ( $sc -gt 3 ) {
      [void] $sc--
      $gaps = @()
      $gaps += ( ( NonContinuous-SubSequence ( 1..$sc ) ) | ForEach-Object {
         $gap1 = ",$_,"
         "0,{0}" -f ( [String]::Join( ',', ( 1..$sc | Where-Object { $gap1 -notmatch "$_," } ) ) )
      } )
      $gaps += 1..( $sc - 1 )
      2..( $sc - 1 ) | ForEach-Object {
         $gap2 = $_ - 1
         $gaps += ( ( SubSequence ( $_..$sc ) ) | ForEach-Object {
            "$gap2,$_"
         } )
      }
      #Write-Host "S $S gaps $gaps"
      $gaps | ForEach-Object {
         $gap3 = ",$_,"
         "$( 0..$sc | Where-Object { $gap3 -notmatch ",$_," } | ForEach-Object {
            $S[$_]
         } )" -replace ' ', ','
      }
   } else { 
      $null
   }
}

( NonContinuous-SubSequence 'a','b','c','d','e' ) | Select-Object length, @{Name='value';Expression={ $_ } } | Sort-Object length, value | ForEach-Object { $_.value }

Prolog

Works with SWI-Prolog.
We explain to Prolog how to build a non continuous subsequence of a list L, then we ask Prolog to fetch all the subsequences. 

% fetch all the subsequences
ncsubs(L, LNCSL) :-
	setof(NCSL, one_ncsubs(L, NCSL), LNCSL).

% how to build one subsequence
one_ncsubs(L, NCSL) :-
	extract_elem(L, NCSL);
	(   sublist(L, L1),
	    one_ncsubs(L1, NCSL)).

% extract one element of the list
% this element is neither the first nor the last.
extract_elem(L, NCSL) :-
	length(L, Len),
	Len1 is Len - 2,
	between(1, Len1, I),
	nth0(I, L, Elem),
	select(Elem, L, NCS1),
	(   NCSL = NCS1; extract_elem(NCS1, NCSL)).

% extract the first or the last element of the list
sublist(L, SL) :-
	(L = [_|SL];
	reverse(L, [_|SL1]),
	reverse(SL1, SL)).

Example : 

?- ncsubs([a,e,i,o,u], L).
L = [[a,e,i,u],[a,e,o],[a,e,o,u],[a,e,u],[a,i],[a,i,o],[a,i,o,u],[a,i,u],[a,o],[a,o,u],[a,u],[e,i,u],[e,o],[e,o,u],[e,u],[i,u]]

Python

Translation of: Scheme
def ncsub(seq, s=0):
    if seq:
        x = seq[:1]
        xs = seq[1:]
        p2 = s % 2
        p1 = not p2
        return [x + ys for ys in ncsub(xs, s + p1)] + ncsub(xs, s + p2)
    else:
        return [[]] if s >= 3 else []
Output:
>>> ncsub(range(1, 4))
[[1, 3]]
>>> ncsub(range(1, 5))
[[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]]
>>> ncsub(range(1, 6))
[[1, 2, 3, 5], [1, 2, 4, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4, 5], [1, 3, 4],
 [1, 3, 5], [1, 3], [1, 4, 5], [1, 4], [1, 5], [2, 3, 5], [2, 4, 5], [2, 4],
 [2, 5], [3, 5]]

A faster Python + Psyco JIT version: 

from sys import argv
import psyco

def C(n, k):
    result = 1
    for d in xrange(1, k+1):
        result *= n
        n -= 1
        result /= d
    return result

# http://oeis.org/A002662
nsubs = lambda n: sum(C(n, k) for k in xrange(3, n+1))

def ncsub(seq):
    n = len(seq)
    result = [None] * nsubs(n)
    pos = 0

    for i in xrange(1, 2 ** n):
        S  = []
        nc = False
        for j in xrange(n + 1):
            k = i >> j
            if k == 0:
                if nc:
                    result[pos] = S
                    pos += 1
                break
            elif k % 2:
                S.append(seq[j])
            elif S:
                nc = True
    return result

from sys import argv
import psyco
psyco.full()
n = 10 if len(argv) < 2 else int(argv[1])
print len( ncsub(range(1, n)) )

Quackery

A sequence of n items has 2^n possible subsequences, including the empty sequence. These correspond to the numbers 0 to 2^n-1, where a one in the binary expansion of the number indicates inclusion in the subsequence of the corresponding item in the sequence. Non-continuous subsequences correspond to numbers where the binary expansion of the number has a one, followed by one or more zeroes, followed by a one. 

  [ dup 1 & dip [ 1 >> ] ] is 2/mod         (   n --> n n )

  [ 0 swap
    [ dup 0 != while
      2/mod iff
        [ dip 1+ ] done
      again ]
    [ dup 0 != while
      2/mod not iff
        [ dip 1+ ] done
      again ]
    [ dup 0 != while
      2/mod iff
        [ dip 1+ ] done
      again ]
    drop 3 = ]             is noncontinuous (   n --> b   )

  [ [] unrot
    [ dup 0 != while
      dip behead tuck
      1 & iff
        [ nested dip rot
          join unrot ]
      else drop
      1 >> again ]
    2drop ]                is bitems        ( [ n --> [   )

  [ [] swap
    dup size bit times
     [ i^ noncontinuous if
        [ dup i^ bitems
          nested rot
          join swap ] ]
     drop ]                is ncsubs        (   [ --> [   )

  ' [ 1 2 3 4 ] ncsubs echo cr 
 
  $ "quackery" ncsubs 72 wrap$
Output:
[ [ 1 3 4 ] [ 1 2 4 ] [ 2 4 ] [ 1 4 ] [ 1 3 ] ]

qackery quckery uckery qckery quakery uakery qakery akery qukery ukery
qkery quacery uacery qacery acery qucery ucery qcery cery quaery uaery
qaery aery query uery qery quackry uackry qackry ackry quckry uckry
qckry ckry quakry uakry qakry akry qukry ukry qkry kry quacry uacry
qacry acry qucry ucry qcry cry quary uary qary ary qury ury qry quackey
uackey qackey ackey quckey uckey qckey ckey quakey uakey qakey akey
qukey ukey qkey key quacey uacey qacey acey qucey ucey qcey cey quaey
uaey qaey aey quey uey qey ey quacky uacky qacky acky qucky ucky qcky
cky quaky uaky qaky aky quky uky qky ky quacy uacy qacy acy qucy ucy qcy
cy quay uay qay ay quy uy qy qacker qucker ucker qcker quaker uaker
qaker aker quker uker qker quacer uacer qacer acer qucer ucer qcer cer
quaer uaer qaer aer quer uer qer quackr uackr qackr ackr quckr uckr qckr
ckr quakr uakr qakr akr qukr ukr qkr kr quacr uacr qacr acr qucr ucr qcr
cr quar uar qar ar qur ur qr qacke qucke ucke qcke quake uake qake ake
quke uke qke quace uace qace ace quce uce qce ce quae uae qae ae que ue
qe qack quck uck qck quak uak qak ak quk uk qk qac quc uc qc qa

R

The idea behind this is to loop over the possible lengths of subsequence, finding all subsequences then discarding those which are continuous. 

ncsub <- function(x)
{
   n <- length(x)
   a <- seq_len(n)
   seqlist <- list()
   for(i in 2:(n-1))
   {
      seqs <- combn(a, i)                                                          # Get all subseqs
      ok <- apply(seqs, 2, function(x) any(diff(x)!=1))                            # Find noncts ones
      newseqs <- unlist(apply(seqs[,ok], 2, function(x) list(x)), recursive=FALSE) # Convert matrix to list of its columns
      seqlist <- c(seqlist, newseqs)                                               # Append to existing list 
   }
   lapply(seqlist, function(index) x[index])
}
# Example usage
ncsub(1:4)
ncsub(letters[1:5])

Racket

Take a simple subsets definition: 

(define (subsets l)
  (if (null? l) '(())
      (append (for/list ([l2 (subsets (cdr l))]) (cons (car l) l2))
              (subsets (cdr l)))))

since the subsets are returned in their original order, it is also a sub-sequences function.

Now add to it a "state" counter which count one for each chunk of items included or excluded. It's always even when we're in an excluded chunk (including the beginning) and odd when we're including items -- increment it whenever we switch from one kind of chunk to the other. This means that we should only include subsequences where the state is 3 (included->excluded->included) or more. Note that this results in code that is similar to the "Generalized monadic filter" entry, except a little simpler. 

#lang racket
(define (non-continuous-subseqs l)
  (let loop ([l l] [x 0])
    (if (null? l) (if (>= x 3) '(()) '())
        (append (for/list ([l2 (loop (cdr l) (if (even? x) (add1 x) x))])
                  (cons (car l) l2))
                (loop (cdr l) (if (odd? x) (add1 x) x))))))
(non-continuous-subseqs '(1 2 3 4))
;; => '((1 2 4) (1 3 4) (1 3) (1 4) (2 4))

Raku

(formerly Perl 6)

Works with: rakudo version 2015-09-24
sub non_continuous_subsequences ( *@list ) {
    @list.combinations.grep: { 1 != all( .[ 0 ^.. .end] Z- .[0 ..^ .end] ) }
}

say non_continuous_subsequences( 1..3 )».gist;
say non_continuous_subsequences( 1..4 )».gist;
say non_continuous_subsequences(   ^4 ).map: {[<a b c d>[.list]].gist};
Output:
((1 3))
((1 3) (1 4) (2 4) (1 2 4) (1 3 4))
([a c] [a d] [b d] [a b d] [a c d])

REXX

This REXX version also works with non-numeric (alphabetic) items   (as well as numbers). 

/*REXX program lists all the  non─continuous subsequences  (NCS),   given a sequence.   */
parse arg list                                   /*obtain optional argument from the CL.*/
if list='' | list==","  then list= 1 2 3 4 5     /*Not specified?  Then use the default.*/
say 'list='  space(list);             say        /*display the list to the terminal.    */
w= words(list)                                   /*W:  is the number of items in list.  */
nums= strip( left(123456789, w) )                /*build a string of decimal digits.    */
tail= right(nums, max(0, w-2) )                  /*construct a fast tail for comparisons*/
#= 0                                             /*#:  number of non─continuous subseq. */
      do j=13  to left(nums,1) || tail           /*step through list (using smart start)*/
      if verify(j, nums) \== 0  then iterate     /*Not one of the chosen  (sequences) ? */
      f= left(j, 1)                              /*use the fist decimal digit of  J.    */
      NCS= 0                                     /*so far, no non─continuous subsequence*/
               do k=2  for length(j)-1           /*search for  "       "          "     */
               x= substr(j, k, 1)                /*extract a single decimal digit of  J.*/
               if x  <= f    then iterate j      /*if next digit ≤, then skip this digit*/
               if x \== f+1  then NCS= 1         /*it's OK as of now  (that is, so far).*/
               f= x                              /*now have a  new  next decimal digit. */
               end   /*k*/
      $=
      if \NCS  then iterate                      /*not OK?  Then skip this number (item)*/
      #= # + 1                                   /*Eureka!  We found a number (or item).*/
               do m=1  for length(j)             /*build a sequence string to display.  */
               $= $ word(list, substr(j, m, 1) ) /*obtain a number (or item) to display.*/
               end   /*m*/

      say 'a non─continuous subsequence: '   $   /*show the non─continuous subsequence. */
      end         /*j*/
say                                              /*help ensure visual fidelity in output*/
if #==0  then #= 'no'                            /*make it look more gooder Angleshy.   */
say  #   "non─continuous subsequence"s(#)    'were found.'             /*handle plurals.*/
exit 0                                           /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
s:  if arg(1)==1  then return '';   return word( arg(2) 's',  1)    /*simple pluralizer.*/
output   when using the input of:     1   2   3   4 
list= 1 2 3 4

a non-continuous subsequence:  1 3
a non-continuous subsequence:  1 4
a non-continuous subsequence:  2 4
a non-continuous subsequence:  1 2 4
a non-continuous subsequence:  1 3 4

5 non-continuous subsequences were found.
output   when using the input of:     a   e   I   o   u 
list= a e I o u

a non-continuous subsequence:  a I
a non-continuous subsequence:  a o
a non-continuous subsequence:  a u
a non-continuous subsequence:  e o
a non-continuous subsequence:  e u
a non-continuous subsequence:  I u
a non-continuous subsequence:  a e o
a non-continuous subsequence:  a e u
a non-continuous subsequence:  a I o
a non-continuous subsequence:  a I u
a non-continuous subsequence:  a o u
a non-continuous subsequence:  e I u
a non-continuous subsequence:  e o u
a non-continuous subsequence:  a e I u
a non-continuous subsequence:  a e o u
a non-continuous subsequence:  a I o u

16 non-continuous subsequences were found.
output   when using the following [five channel Islands (Great Britain)] as input:     Alderney   Guernsey   Herm   Jersey   Sark 
list= Alderney Guernsey Herm Jersey Sark

a non-continuous subsequence:  Alderney Herm
a non-continuous subsequence:  Alderney Jersey
a non-continuous subsequence:  Alderney Sark
a non-continuous subsequence:  Guernsey Jersey
a non-continuous subsequence:  Guernsey Sark
a non-continuous subsequence:  Herm Sark
a non-continuous subsequence:  Alderney Guernsey Jersey
a non-continuous subsequence:  Alderney Guernsey Sark
a non-continuous subsequence:  Alderney Herm Jersey
a non-continuous subsequence:  Alderney Herm Sark
a non-continuous subsequence:  Alderney Jersey Sark
a non-continuous subsequence:  Guernsey Herm Sark
a non-continuous subsequence:  Guernsey Jersey Sark
a non-continuous subsequence:  Alderney Guernsey Herm Sark
a non-continuous subsequence:  Alderney Guernsey Jersey Sark
a non-continuous subsequence:  Alderney Herm Jersey Sark

16 non-continuous subsequences were found.
output   when using the following [six noble gases] as input:     helium   neon   argon   krypton   xenon   radon 
list= helium neon argon krypton xenon radon

a non-continuous subsequence:  helium argon
a non-continuous subsequence:  helium krypton
a non-continuous subsequence:  helium xenon
a non-continuous subsequence:  helium radon
a non-continuous subsequence:  neon krypton
a non-continuous subsequence:  neon xenon
a non-continuous subsequence:  neon radon
a non-continuous subsequence:  argon xenon
a non-continuous subsequence:  argon radon
a non-continuous subsequence:  krypton radon
a non-continuous subsequence:  helium neon krypton
a non-continuous subsequence:  helium neon xenon
a non-continuous subsequence:  helium neon radon
a non-continuous subsequence:  helium argon krypton
a non-continuous subsequence:  helium argon xenon
a non-continuous subsequence:  helium argon radon
a non-continuous subsequence:  helium krypton xenon
a non-continuous subsequence:  helium krypton radon
a non-continuous subsequence:  helium xenon radon
a non-continuous subsequence:  neon argon xenon
a non-continuous subsequence:  neon argon radon
a non-continuous subsequence:  neon krypton xenon
a non-continuous subsequence:  neon krypton radon
a non-continuous subsequence:  neon xenon radon
a non-continuous subsequence:  argon krypton radon
a non-continuous subsequence:  argon xenon radon
a non-continuous subsequence:  helium neon argon xenon
a non-continuous subsequence:  helium neon argon radon
a non-continuous subsequence:  helium neon krypton xenon
a non-continuous subsequence:  helium neon krypton radon
a non-continuous subsequence:  helium neon xenon radon
a non-continuous subsequence:  helium argon krypton xenon
a non-continuous subsequence:  helium argon krypton radon
a non-continuous subsequence:  helium argon xenon radon
a non-continuous subsequence:  helium krypton xenon radon
a non-continuous subsequence:  neon argon krypton radon
a non-continuous subsequence:  neon argon xenon radon
a non-continuous subsequence:  neon krypton xenon radon
a non-continuous subsequence:  helium neon argon krypton radon
a non-continuous subsequence:  helium neon argon xenon radon
a non-continuous subsequence:  helium neon krypton xenon radon
a non-continuous subsequence:  helium argon krypton xenon radon

42 non-continuous subsequences were found.

Ring 

# Project : Non-continuous subsequences

load "stdlib.ring"
list = [1,2,3,4]
items = newlist(pow(2,len(list))-1,len(list))
see "For [1, 2, 3, 4] non-continuous subsequences are:" + nl
powerset(list,4)
showarray(items,4)
see nl

list = [1,2,3,4,5]
items = newlist(pow(2,len(list))-1,len(list))
see "For [1, 2, 3, 4, 5] non-continuous subsequences are:" + nl
powerset(list,5)
showarray(items,5)

func showarray(items,ind)
        for n = 1 to len(items)
             flag = 0
             for m = 1 to ind - 1
                  if items[n][m] = 0 or items[n][m+1] = 0
                     exit
                 ok   
                 if (items[n][m] + 1) != items[n][m+1]
                     flag = 1
                     exit
                 ok
            next
            if flag = 1
               see "[" 
               str = ""
               for x = 1 to len(items[n])
                    if items[n][x] != 0  
                       str = str + items[n][x] + " "
                    ok
               next  
               str = left(str, len(str) - 1)  
               see str + "]" + nl
            ok
        next

func powerset(list,ind)
        num = 0
        num2 = 0
        items = newlist(pow(2,len(list))-1,ind)
        for i = 2 to (2 << len(list)) - 1 step 2
             num2 = 0
             num = num + 1
             for j = 1 to len(list) 
                  if i & (1 << j)
                      num2 = num2 + 1
                      if list[j] != 0
                        items[num][num2] = list[j]
                     ok
                  ok
             next
        next
        return items

Output: 

For [1, 2, 3, 4] non-continuous subsequences are:
[1 3]
[1 4]
[2 4]
[1 2 4]
[1 3 4]

For [1, 2, 3, 4, 5] non-continuous subsequences are:
[1 3]
[1 4]
[2 4]
[1 2 4]
[1 3 4]
[1 5]
[2 5]
[1 2 5]
[3 5]
[1 3 5]
[2 3 5]
[1 2 3 5]
[1 4 5]
[2 4 5]
[1 2 4 5]
[1 3 4 5]

RPL

Translation of: Python
« CASE
     OVER SIZE THEN 
        DUP 2 MOD DUP NOT 4 ROLL DUP 1 1 SUB SWAP TAIL 
        → p2 p1 x xs
        « { p1 p2 } STO+ 
          xs p1 NCSUB 
          IF DUP SIZE THEN 1 « x SWAP + » DOLIST END
          xs p2 NCSUB +
        » END
     3 ≥ THEN 1 →LIST END
  END
» 'NCSUB' STO

{ 1 2 3 4 } 0 NCSUB
{ 1 2 3 4 5 } 0 NCSUB
Output:
2: { { 1 2 4 } { 1 3 4 } { 1 3 } { 1 4 } { 2 4 } }
1: { { 1 2 3 5 } { 1 2 4 5 } { 1 2 4 } { 1 2 5 } { 1 3 4 5 } { 1 3 4 } { 1 3 5 } { 1 3 } { 1 4 5 } { 1 4 } { 1 5 } { 2 3 5 } { 2 4 5 } { 2 4 } { 2 5 } { 3 5 } }

Ruby

Translation of: Tcl

Uses code from Power Set. 

class Array
  def func_power_set
    inject([[]]) { |ps,item|    # for each item in the Array
      ps +                      # take the powerset up to now and add
      ps.map { |e| e + [item] } # it again, with the item appended to each element
    }
  end
  
  def non_continuous_subsequences
    func_power_set.reject {|seq| continuous?(seq)}
  end
  
  def continuous?(seq)
    seq.each_cons(2) {|a, b| return false if a.succ != b}
    true
  end
end

p (1..3).to_a.non_continuous_subsequences
p (1..4).to_a.non_continuous_subsequences
p (1..5).to_a.non_continuous_subsequences
p ("a".."d").to_a.non_continuous_subsequences
Output:
[[1, 3]]
[[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4]]
[[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4], [1, 5], [2, 5], [1, 2, 5], [3, 5], [1, 3, 5], 
 [2, 3, 5], [1, 2, 3, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5], [1, 3, 4, 5]]
[["a", "c"], ["a", "d"], ["b", "d"], ["a", "b", "d"], ["a", "c", "d"]]

It is not the value of the array element and when judging continuation in the position, it changes as follows. 

class Array
  def continuous?(seq)
    seq.each_cons(2) {|a, b| return false if index(a)+1 != index(b)}
    true
  end
end

p %w(a e i o u).non_continuous_subsequences
Output:
[["a", "i"], ["a", "o"], ["e", "o"], ["a", "e", "o"], ["a", "i", "o"], ["a", "u"], ["e", "u"], ["a", "e", "u"], ["i", "u"], ["a", "i", "u"], ["e", "i", "u"], ["a", "e", "i", "u"], ["a", "o", "u"], ["e", "o", "u"], ["a", "e", "o", "u"], ["a", "i", "o", "u"]]

Rust

Translation of: Go
const M: usize = 0;
const C: usize = 1;
const CM: usize = 2;
const CMC: usize = 3;
static SKIP: [usize; 4] = [M, CM, CM, CMC];
static INCL: [usize; 4] = [C, C, CMC, CMC];

fn ncs(s: &Vec<i32>) -> Vec<Vec<i32>> {
    if s.len() < 3 {
        return vec![];
    }
    let mut v1 = n2([].to_vec(), s[1..].to_vec(), M);
    let mut v2 = n2([s[0]].to_vec(), s[1..].to_vec(), C);
    v1.append(&mut v2);
    return v1;
}

fn n2(ss: Vec<i32>, tail: Vec<i32>, seq: usize) -> Vec<Vec<i32>> {
    if tail.len() == 0 {
        if seq != CMC as usize {
            return vec![];
        }
        return [ss].to_vec();
    }
    let mut v1 = n2(ss.clone(), tail[1..].to_vec(), SKIP[seq]);
    let mut v2 = ss.clone();
    v2.push(tail[0]);
    let mut v3 = n2(v2, tail[1..].to_vec(), INCL[seq]);
    v1.append(&mut v3);
    return v1;
}

fn main() {
    let ss = ncs(&[1, 2, 3, 4].to_vec());
    println!("{} non-continuous subsequences:", ss.len());
    for s in ss {
        println!("  {:?}", s);
    }
}
Output:
5 non-continuous subsequences:
  [2, 4]
  [1, 4]
  [1, 3]
  [1, 3, 4]
  [1, 2, 4]

Scala 

object NonContinuousSubSequences extends App {

  private def seqR(s: String, c: String, i: Int, added: Int): Unit = {
    if (i == s.length) {
      if (c.trim.length > added) println(c)
    } else {
      seqR(s, c + s(i), i + 1, added + 1)
      seqR(s, c + " ", i + 1, added)
    }
  }

  seqR("1234", "", 0, 0)
}

Scheme

Translation of: Generalized monadic filter
(define (ncsubseq lst)
  (let recurse ((s 0)
                (lst lst))
    (if (null? lst)
        (if (>= s 3)
            '(())
            '())
        (let ((x (car lst))
              (xs (cdr lst)))
          (if (even? s)
              (append
               (map (lambda (ys) (cons x ys))
                    (recurse (+ s 1) xs))
               (recurse s xs))
              (append
               (map (lambda (ys) (cons x ys))
                    (recurse s xs))
               (recurse (+ s 1) xs)))))))
Output:
> (ncsubseq '(1 2 3))
((1 3))
> (ncsubseq '(1 2 3 4))
((1 2 4) (1 3 4) (1 3) (1 4) (2 4))
> (ncsubseq '(1 2 3 4 5))
((1 2 3 5) (1 2 4 5) (1 2 4) (1 2 5) (1 3 4 5) (1 3 4) (1 3 5) (1 3) (1 4 5) (1 4) (1 5) (2 3 5) (2 4 5) (2 4) (2 5) (3 5))

Seed7 

$ include "seed7_05.s7i";

const func array bitset: ncsub (in bitset: seq, in integer: s) is func
  result
    var array bitset: subseq is 0 times {};
  local
    var bitset: x is {};
    var bitset: xs is {};
    var bitset: ys is {};
  begin
    if seq <> {} then
      x := {min(seq)};
      xs := seq - x;
      for ys range ncsub(xs, s + 1 - s rem 2) do
        subseq &:= x | ys;
      end for;
      subseq &:= ncsub(xs, s + s rem 2);
    elsif s >= 3 then
      subseq &:= {};
    end if;
  end func;

const proc: main is func
  local
    var bitset: seq is {};
  begin
    for seq range ncsub({1, 2, 3, 4}, 0) do
      writeln(seq);
    end for;
  end func;
Output:
{1, 2, 4}
{1, 3, 4}
{1, 3}
{1, 4}
{2, 4}

Sidef

Translation of: Perl
func non_continuous(min, max, subseq=[], has_gap=false) {

    static current = [];

    range(min, max).each { |i|
        current.push(i);
        has_gap && subseq.append([current...]);
        i < max && non_continuous(i.inc, max, subseq, has_gap);
        current.pop;
        has_gap = current.len;
    }

    subseq;
}

say non_continuous(1, 3);
say non_continuous(1, 4);
say non_continuous("a", "d");
Output:
[[1, 3]]
[[1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2, 4]]
[["a", "b", "d"], ["a", "c"], ["a", "c", "d"], ["a", "d"], ["b", "d"]]

Standard ML

Translation of: Generalized monadic filter
fun fence s [] =
      if s >= 3 then
        [[]]
      else
        []

  | fence s (x :: xs) =
      if s mod 2 = 0 then
        map
          (fn ys => x :: ys)
          (fence (s + 1) xs)
        @
          fence s xs
      else
        map
          (fn ys => x :: ys)
          (fence s xs)
        @
          fence (s + 1) xs

fun ncsubseq xs = fence 0 xs
Output:
- ncsubseq [1,2,3];
val it = [[1,3]] : int list list
- ncsubseq [1,2,3,4];
val it = [[1,2,4],[1,3,4],[1,3],[1,4],[2,4]] : int list list
- ncsubseq [1,2,3,4,5];
val it =
  [[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3],
   [1,4,5],[1,4],[1,5],[2,3,5],...] : int list list

Tcl

This Tcl implementation uses the subsets function from Power Set, which is acceptable as that conserves the ordering, as well as a problem-specific test function is_not_continuous and a generic list filter lfilter: 

 proc subsets l {
     set res [list [list]]
     foreach e $l {
         foreach subset $res {lappend res [lappend subset $e]}
     }
     return $res
 }
 proc is_not_continuous seq {
     set last [lindex $seq 0]
     foreach e [lrange $seq 1 end] {
         if {$e-1 != $last} {return 1}
         set last $e
     }
     return 0
 }
 proc lfilter {f list} {
     set res {}
     foreach i $list {if [$f $i] {lappend res $i}}
     return $res
 }

% lfilter is_not_continuous [subsets {1 2 3 4}]
{1 3} {1 4} {2 4} {1 2 4} {1 3 4}

Ursala

To do it the lazy programmer way, apply the powerset library function to the list, which will generate all continuous and non-continuous subsequences of it, and then delete the subsequences that are also substrings (hence continuous) using a judicious combination of the built in substring predicate (K3), negation (Z), and distributing filter (K17) operator suffixes. This function will work on lists of any type. To meet the requirement for structural equivalence, the list items are first uniquely numbered (num), and the numbers are removed afterwards (rSS). 

#import std

noncontinuous = num; ^rlK3ZK17rSS/~& powerset

#show+

examples = noncontinuous 'abcde'
Output:
abce
abd
abde
abe
ac
acd
acde
ace
ad
ade
ae
bce
bd
bde
be
ce

VBScript

Translation of: BBC BASIC
'Non-continuous subsequences - VBScript - 03/02/2021

Function noncontsubseq(l)
    Dim  i, j, g, n, r, s, w, m
    Dim  a, b, c
    n = Ubound(l)
    For s = 0 To n-2
        For g = s+1 To n-1
            a = "["
            For i = s To g-1
                a = a & l(i) & ", "
            Next 'i
            For w = 1 To n-g
                r = n+1-g-w
                For i = 1 To 2^r-1 Step 2
                    b = a
                    For j = 0 To r-1
                        If i And 2^j Then b=b & l(g+w+j) & ", "
                    Next 'j
                    c = (Left(b, Len(b)-1))
                    WScript.Echo Left(c, Len(c)-1) & "]"
					m = m+1
                Next 'i
            Next 'w
        Next 'g
    Next 's
    noncontsubseq = m
End Function 'noncontsubseq

list = Array("1", "2", "3", "4")
WScript.Echo "List: [" & Join(list, ", ") & "]"
nn = noncontsubseq(list)
WScript.Echo nn & " non-continuous subsequences"
Output:
List: [1, 2, 3, 4]
[1, 3]
[1, 3, 4]
[1, 4]
[1, 2, 4]
[2, 4]
5 non-continuous subsequences

Wren

Translation of: Kotlin
Library: Wren-fmt

Needed a bit of doctoring to do the character example as Wren only has strings. 

import "./fmt" for Fmt

var ncs = Fn.new { |a|
    var f = "$d "
    if (a[0] is String) {
        for (i in 0...a.count) a[i] = a[i].bytes[0]
        f = "$c "
    }
    var generate // recursive
    generate = Fn.new { |m, k, c|
        if (k == m) {
            if (c[m - 1] != c[0] + m - 1) {
                for (i in 0...m) Fmt.write(f, a[c[i]])
                System.print()
            }
        } else {
            for (j in 0...a.count) {
                if (k == 0 || j > c[k - 1]) {
                    c[k] = j
                    generate.call(m, k + 1, c)
                }
            }
        }
    }

    for (m in 2...a.count) {
        var c = List.filled(m, 0)
        generate.call(m, 0, c)
    }
}

var a = [1, 2, 3, 4]
ncs.call(a)
System.print()
var ca = ["a", "b", "c", "d", "e"]
ncs.call(ca)
Output:
1 3 
1 4 
2 4 
1 2 4 
1 3 4 

a c 
a d 
a e 
b d 
b e 
c e 
a b d 
a b e 
a c d 
a c e 
a d e 
b c e 
b d e 
a b c e 
a b d e 
a c d e

XPL0

Translation of: Wren
proc NCS(A, Size, Char);
int  A, Size, Char;
int  C, M;

        proc Generate(M, K, C);         \recursive
        int  M, K, C;
        int  I, J;
        [if K = M then
            [if C(M-1) # C(0)+M-1 then
                [for I:= 0 to M-1 do
                    [if Char then ChOut(0, A(C(I)))
                    else IntOut(0, A(C(I)));
                    ChOut(0, ^ );
                    ];
                CrLf(0);
                ];
            ]
        else
            [for J:= 0 to Size-1 do
                [if K = 0 or J > C(K-1) then
                    [C(K):= J;
                    Generate(M, K+1, C);
                    ];
                ];
            ];
        ];

[C:= Reserve(Size*4);
for M:= 2 to Size-1 do Generate(M, 0, C);
];

int  A, CA;
[A:= [1, 2, 3, 4];
NCS(A, 4, false);
CrLf(0);
CA:= [^a, ^b, ^c, ^d, ^e];
NCS(CA, 5, true);
]
Output:
1 3 
1 4 
2 4 
1 2 4 
1 3 4 

a c 
a d 
a e 
b d 
b e 
c e 
a b d 
a b e 
a c d 
a c e 
a d e 
b c e 
b d e 
a b c e 
a b d e 
a c d e

zkl

Translation of: JavaScript
fcn non_continuous_subsequences(ary){
   pwerSet(ary).filter(fcn(list){(not isContinuous(list)) })
}
fcn isContinuous(ary){
   if(ary.len()<2) return(True);
   foreach n in (ary.len()-1){ if(1+ary[n]!=ary[n+1]) return(False); }
   return(True);
}
non_continuous_subsequences(T(1,2,3,4)).println();
fcn pwerSet(list){
  (0).pump(list.len(),List,List,Utils.Helpers.pickNFrom.fp1(list),
     T(T,Void.Write,Void.Write) ) .append(list)
}
fcn brokenSubsequences(str){
   pwerSet(str.split("")).apply("concat")
   .filter('wrap(substr){ (not str.holds(substr)) })
}
brokenSubsequences("1234").println();
Output:
L(L(1,3),L(1,4),L(2,4),L(1,2,4),L(1,3,4))
L("13","14","24","124","134")
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