String Permutations

Skiena Chapter 7

Question 14

Write a function to find all permutations of the letters in a particular string (example: HELLO).

Enumerating

Enumerating - Easy Case

Let's start by counting the total number of permutations of letters in a particular string.

Consider the simplest case, no repeated letters: ABCD

Each position can hold any of the four letters, but we can only choose a letter once. So the total number of permutations of a k-letter string with no repeated letters is $ k! $

Mathematically, we are picking 4 distinct objects, and we are picking all 4 objects at the same time. In other words, 4 pick 4.

This leads to a total number of possible strings:

$ P(4,4) = \dfrac{4!}{(4-4)!} = \dfrac{4!}{1} = 4! $

We are using pick, not choose, because order matters. Including a 4! on the bottom would indicate that order does not matter, and would lead to one possible outcome (obvious - any permutation with the letters ABCD will always be the same set of letters, but in different order, so there is only one outcome).

Enumerating - Multiset Case

Now consider the less trivial case where we have repeated letters: AAABCDD

To count the number of permutations of a multiset, we should use multinomial coefficients. (Note: see AOCP/Multisets and AOCP/Multinomial Coefficients).

Mathematically, this problem is what is known as a multiset problem, where we have a set of letters each occurring with some frequency:

$ \{ a \cdot A, b \cdot B, c \cdot C, d \cdot D \} $

In this case we can write the number of possible permutations by placing each character, one at a time.

Start by placing the A characters. The number of slots to place the A characters is (a) choose (number of slots):

$ \binom{a+b+c+d}{a} $

Once the As have been placed, we can enumerate the B characters The number of slots to place the B characters is (b) choose (number of slots, less A characters):

$ \binom{b+c+d}{b} $

Next, the Cs can be placed, where the number of slots to place the C characters is (c) choose (number of slots, less A and B characters):

$ \binom{c+d}{c} $

and finally, all of the other characters are placed, leaving the Ds with a single remaining configuration:

$ \binom{d}{d} $

Now, the total number of permutations is the product of each of these. The number of configurations of a multiset is given by the multinomial coefficient

$ \binom{n}{a, b, c, d} = \binom{n}{a} \binom{n-a}{b} \binom{n-a-b}{c} \binom{n-a-b-c}{d} = \dfrac{n!}{a! b! c! d!} $

More generally, for a multiset with k objects (denoted a) occurring a certain number of times (denoted r),

$ \{ r_1 \cdot a_1, r_2 \cdot a_2, \dots, r_k \cdot a_k \} $

then if there are n total slots $ r_1 + r_2 + \dots + r_k = n $, the total number of permutations is given by the multinomial coefficient,

$ \binom{n}{r_1, r_2, \dots, r_k} = \binom{n}{r_1} \binom{n-r_1}{r_2} \binom{n - r_1 - r_2}{r_3} \dots $

This can be expressed more concisely in terms of factorials:

$ \binom{n}{r_1, r_2, \dots, r_k} = \dfrac{n!}{r_1! r_2! \dots r_k!} $

Enumerating - Infinite Letters Case

Suppose we have an infinite pool of characters, and we wish to know the number of words of a given length that can be formed using these infinite pools of letters.

We can still treat this as a multiset problem, but this time the number of characters is infinite. This multiset is denoted:

$ \{ \infty \cdot A, \infty \cdot B, \infty \cdot C, \infty \cdot D \} $

In this case, the number of permutations of length n that can be formed from the 4 different characters is given by the simple expression,

$ n^4 $

More generally, if we have an infinite multiset with k distinct elements,

$ \{ \infty \cdot r_1, \infty \cdot r_2, \dots, \infty \cdot r_k \} $

then the number of words of length n that can be formed is given by:

$ n^k $

Enumerating - Stars and Bars

The stars and bars theorem/approach is another way to look at assembling/enumerating permutations.

If we have n objects being placed into k partitions, we can think of this as placing k-1 bars among the n objects.

In this case, the total number of slots for objects + bars is n + k - 1, and we are placing k - 1 bars, so we have the total number of outcomes given by

$ \binom{n+k-1}{k-1} $

Generating

The task of actually generating all of the permutations of words that contain a certain set of characters can be reduced to a simpler task, as can most permutation-generating tasks.