Stirling numbers of the second kind: Difference between revisions
Stirling numbers of the second kind (view source)
Revision as of 14:35, 25 March 2024
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Line 39:
{{trans|Python}}
<
F sterling2(n, k)
Line 77:
E
print("#.\n(#. digits, k = #.)\n".format(previous, String(previous).len, k - 1))
L.break</
{{out}}
Line 104:
{{works with|ALGOL 68G|Any - tested with release 2.8.3.win32}}
Uses the LONG LONG INT mode of Algol 68g which allows large precision integers. As the default precision of LONG LONG INT is too small, the precision is specified via a pragmatic comment.
<
# specify the precision of LONG LONG INT, somewhat under 160 digits are #
Line 132:
print( ( "Stirling numbers of the second kind:", newline ) );
print( ( " k" ) );
FOR k FROM 0 TO max stirling DO print( ( whole( k, -
print( ( newline, " n", newline ) );
FOR n FROM 0 TO max stirling DO
print( ( whole( n, -2 ) ) );
FOR k FROM 0 TO n DO
print( ( whole( s2[ n, k ], -
OD;
print( ( newline ) )
Line 153:
print( ( whole( max 100, 0 ), newline ) )
END
END
</syntaxhighlight>
{{out}}
<pre>
Stirling numbers of the second kind:
k
n
0
1
2
3
4
5
6
7
8
9
10
11
12
Maximum Stirling number of the second kind with n = 100:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
Line 177 ⟶ 178:
=={{header|ALGOL W}}==
<
integer MAX_STIRLING;
MAX_STIRLING := 12;
Line 195 ⟶ 196:
write( "Stirling numbers of the second kind:" );
write( " k" );
for k := 0 until MAX_STIRLING do writeon( i_w :=
write( " n" );
for n := 0 until MAX_STIRLING do begin
write( i_w := 2, s_w := 0, n );
for k := 0 until n do writeon( i_w :=
end for_n
end
end.
</syntaxhighlight>
{{out}}
<pre>
Stirling numbers of the second kind:
k
n
0
1
2
3
4
5
6
7
8
9
10
11
12
</pre>
=={{header|BASIC}}==
<
20 DIM S2(12,12),F(12)
30 FOR N=0 TO 12: READ F(N): NEXT N
Line 239 ⟶ 241:
140 NEXT N
150 DATA ##,##,#####,######,#######,########
160 DATA ########,#######,#######,######,#####,###,##</
{{out}}
<pre> 1
Line 256 ⟶ 258:
=={{header|C}}==
<
#include <stdio.h>
#include <stdlib.h>
Line 317 ⟶ 319:
stirling_cache_destroy(&sc);
return 0;
}</
{{out}}
Line 340 ⟶ 342:
=={{header|C++}}==
{{libheader|GMP}}
<
#include <iomanip>
#include <iostream>
Line 392 ⟶ 394:
std::cout << max << '\n';
return 0;
}</
{{out}}
Line 417 ⟶ 419:
=={{header|D}}==
{{trans|Java}}
<
import std.conv;
import std.functional;
Line 471 ⟶ 473:
}
}
}</
{{out}}
<pre>Stirling numbers of the second kind:
Line 491 ⟶ 493:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
(115 digits, k = 28)</pre>
=={{header|EasyLang}}==
<syntaxhighlight lang="easylang">
print "Unsigned Stirling numbers of the second kind:"
len a[] 13 ; arrbase a[] 0
len b[] 13 ; arrbase b[] 0
a[0] = 1
print 1
for n = 1 to 12
b[0] = 0
write 0 & " "
for k = 1 to n - 1
b[k] = k * a[k] + a[k - 1]
write b[k] & " "
.
b[n] = 1
write 1 & " "
print ""
swap a[] b[]
.
</syntaxhighlight>
=={{header|Factor}}==
{{works with|Factor|0.99 development version 2019-07-10}}
<
math.extras prettyprint sequences ;
RENAME: stirling math.extras => (stirling)
Line 514 ⟶ 537:
"Maximum value from the 100 _ stirling row:" print
100 <iota> [ 100 swap stirling ] map supremum .</
{{out}}
<pre>
Line 538 ⟶ 561:
=={{header|FreeBASIC}}==
<
dim as ubyte n, k
dim as string outstr
Line 569 ⟶ 592:
next k
print outstr
next n</
<pre>Stirling numbers of the second kind
Line 590 ⟶ 613:
=={{header|Fōrmulæ}}==
'''Solution'''
'''Version 1. Recursive'''
[[File:Fōrmulæ - Stirling numbers of the second kind 01.png]]
'''Test case 1. Show the Stirling numbers of the second kind, S₂(n, k), up to S₂(12, 12)'''
[[File:Fōrmulæ - Stirling numbers of the second kind 02.png]]
[[File:Fōrmulæ - Stirling numbers of the second kind 03.png]]
'''Version 2. Non recursive'''
A faster, non recursive version is presented. This constructs a matrix.
[[File:Fōrmulæ - Stirling numbers of the second kind 04.png]]
'''Test case 1. Show the Stirling numbers of the second kind, S₂(n, k), up to S₂(12, 12)'''
[[File:Fōrmulæ - Stirling numbers of the second kind 05.png]]
(the result is the same as recursive version)
'''Test case 2. Find the maximum value of S₂(n, k) where n ≤ 100'''
[[File:Fōrmulæ - Stirling numbers of the second kind 06.png]]
[[File:Fōrmulæ - Stirling numbers of the second kind 07.png]]
=={{header|Go}}==
<
import (
Line 649 ⟶ 698:
fmt.Println(max)
fmt.Printf("which has %d digits.\n", len(max.String()))
}</
{{out}}
Line 675 ⟶ 724:
</pre>
=={{header|Haskell}}==
<
import Data.List (groupBy)
import qualified Data.MemoCombinators as Memo
Line 700 ⟶ 749:
where
table :: [[(Int, Int)]]
table = groupBy (\a b -> fst a == fst b) $ (,) <$> [0..12] <*> [0..12]</
{{out}}
<pre>n/k 0 1 2 3 4 5 6 7 8 9 10 11 12
Line 759 ⟶ 808:
=={{header|Java}}==
<
import java.math.BigInteger;
import java.util.HashMap;
Line 820 ⟶ 869:
}
</syntaxhighlight>
{{out}}
Line 840 ⟶ 889:
12 0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1
The maximum value of S2(100, k) =
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
(115 digits, k = 28)
</pre>
=={{header|jq}}==
'''Adapted from [[#Wren|Wren]]'''
'''Works with gojq, the Go implementation of jq'''
The following program is more complex than it otherwise would be because
it ensures that the cache of values computed in the first part
is available in the second part.
<syntaxhighlight lang="jq">def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# Input: {computed} (the cache)
# Output: {computed, result}
def stirling2($n; $k):
"\($n),\($k)" as $key
| if .computed[$key] then .result = .computed[$key]
elif ($n == 0 and $k == 0) then .result = 1
elif (($n > 0 and $k == 0) or ($n == 0 and $k > 0)) then .result = 0
elif ($k == $n) then .result = 1
elif ($k > $n) then .result = 0
else stirling2($n-1; $k-1) as $s1
| ($s1 | stirling2($n-1; $k)) as $s2
| ($s1.result + ($k * $s2.result)) as $result
| $s2
| .computed[$key] = $result
| .result = $result
end ;
# Set .emit to a table of values and .computed to a cache of stirling2 values
# Output: {emit, computed}
def part1(max):
{emit: "Unsigned Stirling numbers of the second kind:"}
| .emit += "\n" + "n/k" +
([range(0; max+1) | lpad(10)] | join(""))
| reduce range(0; max+1) as $n ( .computed = {};
.emit += "\n" + ($n | lpad(3))
| reduce range(0; $n+1) as $k (.;
stirling2($n; $k)
| .emit += (.result | lpad(10) ) )) ;
# "The maximum value of S2($m, k) ..."
# Input: {computed}
def part2($m):
"The maximum value of S2(\($m), k) =",
first(
foreach range(1;$m+1) as $k ({computed:{}, previous: 0};
stirling2($m; $k) as $current
| if ($current.result > .previous)
then .previous = $current.result
else
.emit = "\(.previous)\n(\(.previous|tostring|length) digits, k = \($k - 1))"
end;
select(.emit).emit) );
part1(12)
| .emit,
part2(100)
</syntaxhighlight>
{{out}}
<pre>
Unsigned Stirling numbers of the second kind:
n/k 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1
1 0 1
2 0 1 1
3 0 1 3 1
4 0 1 7 6 1
5 0 1 15 25 10 1
6 0 1 31 90 65 15 1
7 0 1 63 301 350 140 21 1
8 0 1 127 966 1701 1050 266 28 1
9 0 1 255 3025 7770 6951 2646 462 36 1
10 0 1 511 9330 34105 42525 22827 5880 750 45 1
11 0 1 1023 28501 145750 246730 179487 63987 11880 1155 55 1
12 0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1
The maximum value of S2(100, k) =
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
(115 digits, k = 28)
Line 845 ⟶ 975:
=={{header|Julia}}==
<
const s2cache = Dict()
Line 883 ⟶ 1,013:
println("\nThe maximum for stirling2(100, _) is: ", maximum(k-> stirlings2(BigInt(100), BigInt(k)), 1:100))
</
<pre>
0 1 2 3 4 5 6 7 8 9 10 11 12
Line 906 ⟶ 1,036:
=={{header|Kotlin}}==
{{trans|Java}}
<
fun main() {
Line 958 ⟶ 1,088:
COMPUTED[key] = result
return result
}</
{{out}}
<pre>Stirling numbers of the second kind:
Line 978 ⟶ 1,108:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
(115 digits, k = 28)</pre>
=={{header|Lua}}==
<syntaxhighlight lang="lua">
do -- show some Stirling numbers of the second kind
local MAX_STIRLING = 12;
-- construct a matrix of Stirling numbers up to max n, max n
local s2 = {}
for n = 0, MAX_STIRLING do
s2[ n ] = {}
for k = 0, MAX_STIRLING do s2[ n ][ k ] = 0 end
end
for n = 0, MAX_STIRLING do s2[ n ][ n ] = 1 end
for n = 0, MAX_STIRLING - 1 do
for k = 1, n do
s2[ n + 1 ][ k ] = k * s2[ n ][ k ] + s2[ n ][ k - 1 ]
end
end
io.write( "Stirling numbers of the second kind:\n" )
io.write( " k" )
for k = 0, MAX_STIRLING do io.write( string.format( "%8d", k ) ) end
io.write( "\n" )
io.write( " n\n" );
for n = 0, MAX_STIRLING do
io.write( string.format( "%2d", n ) )
for k = 0, n do io.write( string.format( "%8d", s2[ n ][ k ] ) ) end
io.write( "\n" )
end
end
</syntaxhighlight>
{{out}}
<pre>
Stirling numbers of the second kind:
k 0 1 2 3 4 5 6 7 8 9 10 11 12
n
0 1
1 0 1
2 0 1 1
3 0 1 3 1
4 0 1 7 6 1
5 0 1 15 25 10 1
6 0 1 31 90 65 15 1
7 0 1 63 301 350 140 21 1
8 0 1 127 966 1701 1050 266 28 1
9 0 1 255 3025 7770 6951 2646 462 36 1
10 0 1 511 9330 34105 42525 22827 5880 750 45 1
11 0 1 1023 28501 145750 246730 179487 63987 11880 1155 55 1
12 0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1
</pre>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<
Max[Abs[StirlingS2[100, #]] & /@ Range[0, 100]]</
{{out}}
<pre> k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10 k=11 k=12
Line 1,001 ⟶ 1,179:
=={{header|Nim}}==
As for Stirling numbers of first kind, a simple program using recursive definition is enough to solve the task when not using big numbers.
<
proc s2(n, k: Natural): Natural =
Line 1,014 ⟶ 1,192:
for k in 0..n:
stdout.write ($s2(n, k)).align(8)
stdout.write '\n'</
{{out}}
Line 1,035 ⟶ 1,213:
Now, to solve the second part of the task, we have to use big numbers. As for Stirling numbers of first kind, we also use a cache to avoid to repeat multiple times the same computations.
{{libheader|bignum}}
<
import bignum
Line 1,054 ⟶ 1,232:
echo "Maximum Stirling number of the second kind with n = 100:"
echo max</
{{out}}
Line 1,061 ⟶ 1,239:
=={{header|Perl}}==
<
use warnings;
use bigint;
Line 1,092 ⟶ 1,270:
say "\nMaximum value from the S2(100, *) row:";
say max map { Stirling2(101,$_) } 0..100;</
{{out}}
<pre>Unsigned Stirling2 numbers of the second kind: S2(n, k):
Line 1,116 ⟶ 1,294:
{{libheader|Phix/mpfr}}
{{trans|Go}}
<!--<
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
Line 1,155 ⟶ 1,333:
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nThe maximum S2(100,k): %s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m100</span><span style="color: #0000FF;">)))</span>
<!--</
{{out}}
<pre>
Line 1,180 ⟶ 1,358:
=={{header|Prolog}}==
{{works with|SWI Prolog}}
<
stirling2(N, N, 1):-!.
Line 1,221 ⟶ 1,399:
writeln('Maximum value of S2(n,k) where n = 100:'),
max_stirling2(100, M),
writeln(M).</
{{out}}
Line 1,244 ⟶ 1,422:
=={{header|Python}}==
{{trans|Java}}
<syntaxhighlight lang="python">
computed = {}
Line 1,284 ⟶ 1,462:
print("{0}\n({1} digits, k = {2})\n".format(previous, len(str(previous)), k - 1))
break
</syntaxhighlight>
{{out}}
<pre>
Line 1,309 ⟶ 1,487:
=={{header|Quackery}}==
<
over size -
space swap of
Line 1,340 ⟶ 1,518:
0 100 times
[ 100 i^ 1+ s2 max ]
echo cr</
{{out}}
Line 1,365 ⟶ 1,543:
{{works with|Rakudo|2019.07.1}}
<syntaxhighlight lang="raku"
((1,), { (0, |@^last) »+« (|(@^last »*« @^last.keys), 0) } … *)[n;k]
}
Line 1,382 ⟶ 1,560:
say "\nMaximum value from the S2(100, *) row:";
say (^100).map( { Stirling2 100, $_ } ).max;</
{{out}}
<pre>Stirling numbers of the second kind: S2(n, k):
Line 1,405 ⟶ 1,583:
=={{header|REXX}}==
Some extra code was added to minimize the displaying of the column widths.
<syntaxhighlight lang="text">/*REXX program to compute and display Stirling numbers of the second kind. */
parse arg lim . /*obtain optional argument from the CL.*/
if lim=='' | lim=="," then lim= 12 /*Not specified? Then use the default.*/
Line 1,442 ⟶ 1,620:
end /*c*/
say center(r, wgi) strip( substr($,2), 'T') /*display a single row of the grid. */
end /*r*/ /*stick a fork in it, we're all done. */</
{{out|output|text= when using the default input:}}
<pre>
Line 1,464 ⟶ 1,642:
The maximum value (which has 115 decimal digits):
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
</pre>
=={{header|RPL}}==
{{works with|RPL|HP49-C}}
« '''IF''' DUP2 AND NOT '''THEN''' ==
'''ELSE'''
SWAP 1 - OVER
DUP2 1 - <span style="color:blue">S2</span> 4 ROLLD <span style="color:blue">S2</span> * +
'''END'''
» '<span style="color:blue">S2</span>' STO <span style="color:grey">''@ ( n k → S2(n,k) )''</span>
12 12 « <span style="color:blue">S2</span> » LCXM
{{out}}
<pre>
1: [[ 1 0 0 0 0 0 0 0 0 0 0 0 ]
[ 1 1 0 0 0 0 0 0 0 0 0 0 ]
[ 1 3 1 0 0 0 0 0 0 0 0 0 ]
[ 1 7 6 1 0 0 0 0 0 0 0 0 ]
[ 1 15 25 10 1 0 0 0 0 0 0 0 ]
[ 1 31 90 65 15 1 0 0 0 0 0 0 ]
[ 1 63 301 350 140 21 1 0 0 0 0 0 ]
[ 1 127 966 1701 1050 266 28 1 0 0 0 0 ]
[ 1 255 3025 7770 6951 2646 462 36 1 0 0 0 ]
[ 1 511 9330 34105 42525 22827 5880 750 45 1 0 0 ]
[ 1 1023 28501 145750 246730 179487 63987 11880 1155 55 1 0 ]
[ 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1 ]]
</pre>
=={{header|Ruby}}==
<
def sterling2(n, k)
Line 1,492 ⟶ 1,696:
puts "\nMaximum value from the sterling2(100, k)";
puts (1..100).map{|a| sterling2(100,a)}.max
</syntaxhighlight>
{{out}}<pre>Sterling2 numbers:
n/k 0 1 2 3 4 5 6 7 8 9 10 11 12
Line 1,513 ⟶ 1,717:
</pre>
=={{header|Sidef}}==
<
stirling2(n, k)
}
Line 1,534 ⟶ 1,739:
say "\nMaximum value from the S2(#{n}, *) row:"
say { S2(n, _) }.map(^n).max
}</
{{out}}
<pre>
Line 1,557 ⟶ 1,762:
Alternatively, the '''S2(n,k)''' function can be defined as:
<
func S2(_, (0)) { 0 }
func S2((0), _) { 0 }
func S2(n, k) is cached { S2(n-1, k)*k + S2(n-1, k-1) }</
=={{header|Tcl}}==
{{trans|Java}}
<
set nk [list $n $k]
if {[info exists ::S2cache($nk)]} {
Line 1,614 ⟶ 1,819:
}
}
main</
{{out}}
<pre>
Line 1,639 ⟶ 1,844:
=={{header|Visual Basic .NET}}==
{{trans|Java}}
<
Module Module1
Line 1,708 ⟶ 1,913:
End Sub
End Module</
{{out}}
<pre>Stirling numbers of the second kind:
Line 1,733 ⟶ 1,938:
{{libheader|Wren-big}}
{{libheader|Wren-fmt}}
<
import "./fmt" for Fmt
var computed = {}
Line 1,771 ⟶ 1,976:
break
}
}</
{{out}}
Line 1,796 ⟶ 2,001:
=={{header|zkl}}==
<
var seen=Dictionary(); // cache for recursion
if(n==k) return(1); // (0.0)==1
Line 1,804 ⟶ 2,009:
if(Void==(s2 := seen.find(z2))){ s2 = seen[z2] = stirling2(n-1,k-1) }
k*s1 + s2; // k is first to cast to BigInt (if using BigInts)
}</
<
N,mx := 12, [1..N].apply(fcn(n){ [1..n].apply(stirling2.fp(n)) }).flatten() : (0).max(_);
fmt:="%%%dd".fmt("%d".fmt(mx.numDigits + 1)).fmt; // "%9d".fmt
Line 1,812 ⟶ 2,017:
foreach row in ([0..N]){
println("%3d".fmt(row), [0..row].pump(String, stirling2.fp(row), fmt));
}</
{{out}}
<pre style="font-size:83%">
Line 1,832 ⟶ 2,037:
</pre>
{{libheader|GMP}} GNU Multiple Precision Arithmetic Library
<
N=100;
S2100:=[BI(2)..N].apply(stirling2.fp(BI(N))).reduce(fcn(m,n){ m.max(n) });
println("Maximum value from the S2(%d,*) row (%d digits):".fmt(N,S2100.numDigits));
println(S2100);</
{{out}}
<pre style="font-size:83%">
|