# Talk:Factorial base numbers indexing permutations of a collection

## Mojibake and misspellings

(As the task description is fairly difficult to read as it stands, I took a shot at revising it slightly, correcting misspellings and translating the erroneous Windows-1252->utf8 mojibake. Note that this is just my interpretation as I understood it. I am not the task author. Feel free to edit / correct anything I have gotten wrong.)

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You need a random arrangement of a deck of cards, you are sick of lame ways of doing this.

This task is a super-cool way of doing this using factorial base numbers.

The first 25 factorial base numbers in increasing order are:

0.0.0, 0.0.1, 0.1.0, 0.1.1, 0.2.0, 0.2.1, 1.0.0, 1.0.1, 1.1.0, 1.1.1, 1.2.0, 1.2.1, 2.0.0, 2.0.1, 2.1.0, 2.1.1, 2.2.0, 2.2.1, 3.0.0, 3.0.1, 3.1.0, 3.1.1, 3.2.0, 3.2.1, 1.0.0.0

Observe that the least significant digit is base 2, the next; base 3, and so on. In general, an **n**-digit factorial base number will use the digits **0..n** in base **n+1** for each place **1..n** (from least to most significant.)

It is simple to produce a 1-to-1 mapping between an **n** digit factorial base number and permutations of an **n+1** element array:

0.0.0 -> 0123 0.0.1 -> 0132 0.1.0 -> 0213 0.1.1 -> 0231 0.2.0 -> 0312 0.2.1 -> 0321 1.0.0 -> 1023 1.0.1 -> 1032 1.1.0 -> 1203 1.1.1 -> 1230 1.2.0 -> 1302 1.2.1 -> 1320 2.0.0 -> 2013 2.0.1 -> 2031 2.1.0 -> 2103 2.1.1 -> 2130 2.2.0 -> 2301 2.2.1 -> 2310 3.0.0 -> 3012 3.0.1 -> 3021 3.1.0 -> 3102 3.1.1 -> 3120 3.2.0 -> 3201 3.2.1 -> 3210

The following pseudo-code demonstrates the procedure to generate the mapping:

Starting with **m=0** and **Ω**, an array of elements to be permutated, for each digit **g**
starting with the **most** significant digit in the factorial base number.

- If
**g**is greater than zero, rotate the elements from**m**to**m+g**in**Ω**(see example) - Increment
**m**and repeat the first step using the next most significant digit until the factorial base number is exhausted.

For example: using the factorial base number **2.0.1** and **Ω** = **0 1 2 3** where place 0 in both is the most significant (left-most) digit/element.

- Step 1:
**m=0 g=2**; Rotate places 0 through 2.**0 1 2 3**becomes**2 0 1 3** - Step 2:
**m=1 g=0**; No action. - Step 3:
**m=2 g=1**; Rotate places 2 through 3.**2 0 1 3**becomes**2 0 3 1**

- Task

**Part 1:**Use your function to recreate the permutation table of 3 digit factorial base numbers as above. Show the output here.

**Part 2:**Use your function to generate all permutations of 11 digits. Count them, don't display them, and compare this method with the method in RCs Permutations task.

**Part 3:**Use your function to create the corresponding permutations of the following two random 51 digit factorial base numbers:

39.49.7.47.29.30.2.12.10.3.29.37.33.17.12.31.29.34.17.25.2.4.25.4.1.14.20.6.21.18.1.1.1.4.0.5.15.12.4.3.10.10.9.1.6.5.5.3.0.0.0 51.48.16.22.3.0.19.34.29.1.36.30.12.32.12.29.30.26.14.21.8.12.1.3.10.4.7.17.6.21.8.12.15.15.13.15.7.3.12.11.9.5.5.6.6.3.4.0.3.2.1

Use the following shoe of cards as your **Ω** array:

A♠K♠Q♠J♠10♠9♠8♠7♠6♠5♠4♠3♠2♠A♥K♥Q♥J♥10♥9♥8♥7♥6♥5♥4♥3♥2♥A♦K♦Q♦J♦10♦9♦8♦7♦6♦5♦4♦3♦2♦A♣K♣Q♣J♣10♣9♣8♣7♣6♣5♣4♣3♣2♣

**Part 4:**Finally, create your own 51 digit factorial base number and produce the corresponding permutation of the above shoe.

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- Thanks--Nigel Galloway (talk) 11:21, 10 December 2018 (UTC)

## Why not using the inverse of Knuth_shuffle

this has the same factorial base number, but one has to do only length-1 swaps instead of rotation. It ist easy to create a map between source and goal arrangement.

source 1.3.2.0 with value,indices (0,3|1,0|2,1|3,1) goal 0,1,2,3 with value,indices (0,0|1,1|2,2|3,3) start index = 0 first swap 0 <-> 1 => index 0 <-> 3 -> update Value,Indices for swapped values -> (Swap Index = 3 out of [0..3]) source 0.3.2.1 with value,indices (0,0|1,3|2,2|3,1) index++ => 1 swap 3 <-> 1 => index 1 <-> 3 (Swap Index = 3 out of [index..3]) source 0.1.2.3 with value,indices (0,0|1,1|2,2|3,3) index++ => 2 index 2 = index of value -> no swap; source 0.1.2.3 with value,indices (0,0|1,1|2,2|3,3) index++ => 3 index 3 = index of value -> no swap; source 0.1.2.3 with value,indices (0,0|1,1|2,2|3,3)

I see that the advantage of the rotation is the lexicographical order of generating the permutations, but what is it good for?

- see First perfect square in base N with N unique digits#Using Factorial base numbers indexing permutations of a collection--Nigel Galloway (talk) 18:34, 30 May 2019 (UTC)