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title TARGET DECK FILE TAGS tags
Permutations Obsidian::STEM combinatorics set
combinatorics
set

Overview

A permutation of some n objects is a (possible) rearrangement of those n objects. The number of permutations is n! since there are n possible ways to pick the first object, (n - 1) possible ways to pick the second, and so on.

void next(const size_t n, int A[static n]) {
  size_t pivot = -1;
  for (size_t i = n - 1; i >= 1; --i) {
    if (A[i - 1] < A[i]) {
      pivot = i - 1;
      break;
    }
  }
  if (pivot == -1) {
    reverse(0, n - 1, A);
    return;
  }
  size_t j = pivot;
  for (size_t i = pivot + 1; i < n; ++i) {
    if (A[pivot] < A[i] && (j == pivot || A[i] < A[j])) {
      j = i;
    }
  }
  swap(pivot, j, A);
  reverse(pivot + 1, n - 1, A);
}

void permutations(const size_t n, int A[static n]) {
  size_t iters = factorial(n);
  for (size_t i = 0; i < iters; ++i) {
    print_array(n, A);
    next(n, A);
  }
}

The above approach prints out all permutations of an array (assuming distinct values).

%%ANKI Basic What is a permutation? Back: An ordered arrangement of some collection of objects. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How many permutations are there of n objects? Back: n! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How is n! written recursively? Back: n(n - 1)! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How is n! permutations of n objects derived? Back: There are n choices for the first position, n - 1 choices for the second, etc. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What combinatorial concept explains n! permutations of n objects? Back: The multiplicative principle. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How does sorting relate to the concept of permutations? Back: Sorting aims to efficiently find a specific permutation. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf. Tags: algorithm

END%%

%%ANKI Basic What symbol denotes "n factorial"? Back: n! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic n! is shorthand for what other "big operator" formula? Back: \Pi_{k=1}^n k Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is the identity element of \cdot (multiplication)? Back: 1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does 0! (factorial) evaluate to? Back: 1 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Why might 0! = 1 (barring convention)? Back: Because the empty product is 1, the multiplication identity. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What combinatorial concept explains the number of bijective functions between two finite sets? Back: Permutations (factorials). Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How many bijective functions exist between \{1, 2, 3\} and \{a, b, c\}? Back: 3! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How many bijective functions exist between finite sets A and B where |A| = |B| = n? Back: n! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

Lexicographic Ordering

We can find the next lexicographic ordering of an array via a procedure of "pivot", "swap", and "reverse". The function void next(const size_t n, int A[static n]) defined in #Overview shows the details, taking in a permutation and producing the next, lexicographically speaking. To prove correctness, consider the following:

[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]

Here the RHS side is the longest increasing sequence we could find, from right to left. That is, a_{i+1} > a_{i+2} > \cdots > a_n. Denote a_i as the pivot. Next, swap the smallest element in the RHS greater than a_i, say a_j, with a_i. This produces

[ a₁ a₂ ... aⱼ | aᵢ₊₁ aᵢ₊₂ ... aᵢ ... aₙ ]

Notice the RHS remains in sorted order. Since a_j was the next smallest element, reversing the reverse-sorted RHS produces the next permutation, lexicographically speaking:

[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]

Eventually the swapped a_j will be the largest in the RHS ensuring that the breakpoint will eventually move one more position leftward.

%%ANKI Basic What algorithm does NEXT_LEXICO_ARRAY refer to? Back: The finding of the next lexicographic ordering of an array. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What does array A's next lexicographic ordering refer to? Back: The permutation that follows A in a sorted list of all distinct permutations of A. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic How does lexicographic ordering of arrays relate to permutations of an array? Back: Each lexicographic ordering corresponds to a permutation. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: How many invocations guarantee all permutations of A[1:n]? Back: n! Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: When does < n! iterations yield all permutations of A[1:n]? Back: When A contains duplicates. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is NEXT_LEXICO_ARRAY's worst-case running time? Back: O(n) Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is NEXT_LEXICO_ARRAY's best-case running time? Back: \Omega(n) Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is NEXT_LEXICO_ARRAY's auxiliary memory usage? Back: O(1) Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the next lexicographic ordering of [ 1 3 2 4 ]? Back: [ 1 3 4 2 ] Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the next lexicographic ordering of [ 2 1 4 3 ]? Back: [ 2 3 1 4 ] Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the next lexicographic ordering of [ 4 3 2 1 ]? Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the output of NEXT_LEXICO_ARRAY([ 1 2 3 4 ])? Back: [ 1 2 4 3 ] Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the output of NEXT_LEXICO_ARRAY([ 4 3 2 1 ])? Back: [ 1 2 3 4 ] Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: Which element will be the pivot of [ 1 2 3 4 ]? Back: 3 Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: Which element will be the pivot of [ 4 3 2 1 ]? Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What property does the RHS of A[1:n] exhibit before swapping?

[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]

Back: Values are in non-increasing order. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What property does the LHS of A[1:n] exhibit before swapping?

[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]

Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What property does the RHS of A[1:n] exhibit after swapping?

[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]

Back: Values are in non-increasing order. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What property does the LHS of A[1:n] exhibit after swapping?

[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]

Back: N/A Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic What is the first step taken in the NEXT_LEXICO_ARRAY algorithm? Back: Finding the pivot element. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What does the "pivot" refer to? Back: The element preceding the longest increasing subarray from right-to-left. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Cloze The NEXT_LEXICO_ARRAY algorithm can be summed up as "{pivot}", "{swap}", and "{reverse}". Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: Which element is swapped with the pivot? Back: The smallest element to its right that is greater than it. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What is done after swapping the pivot element? Back: Reverse the subarray to the right of where the pivot element was originally located. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What step follows swapping A[1:n]'s pivot element?

[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]

Back: Reverse the elements to the right of a_j. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

%%ANKI Basic NEXT_LEXICO_ARRAY: What invariant is maintained after swapping the pivot? Back: The elements to the right of the original pivot remain in non-increasing order. Reference: https://leetcode.com/problems/next-permutation/description/

END%%

Falling Factorials

If we generalize to choosing k \leq n elements of n objects, we can calculate the k-permutation of n. This is denoted as (n)_k, sometimes called the falling factorial. (n)_k = \frac{n!}{(n - k)!}

The derivation works by noting that we have n - 0 possible ways to pick the first object, n - 1 ways to pick the second, up until n - (k - 1) ways to pick the last object.

%%ANKI Basic What is a k-permutation? Back: An ordered arrangement, containing k elements, of some collection of objects. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is the closed formula for falling factorial (n)_k? Back: (n)_k = \frac{n!}{(n - k)!} Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How is the number of k-permutations of n objects denoted? Back: (n)_k Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How is n! written equivalently as a falling factorial? Back: (n)_n Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How is (n)_k k-permutations of n objects derived? Back: There are n choices for the first position, n - 1 choices for the second, etc. up until n - (k - 1) choices for the last position. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What does (n)_n evaluate to? Back: n! Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What does (n)_0 evaluate to? Back: 1 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What combinatorial problem does (n)_0 represent? Back: The number of ways to choose 0 objects from n choices. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Cloze In a k-permutation of n objects, there are n - 0 choices for first object and {n - (k - 1)} choices for the last object. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What combinatorial concept explains the number of injective functions between two finite sets? Back: k-permutations (falling factorials). Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic How many injective functions exist between \{1, 2, 3\} and \{a, b, c, d, e\}? Back: (5)_3 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

Bibliography