### 48.2 Functions and Variables for Combinatorics

Function: apply_cycles (cl,l)

  Permutes the list or set l applying to it the list of cycles cl. The cycles at the end of the list are applied first and the first cycle in the list cl is the last one to be applied.

  See also `permute`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) lis1:[a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                              2        x  1
      (%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                       y  2

      (%i3) apply_cycles ([[1, 6], [2, 6, 5, 7]], lis1);
                        x  1                       2
      (%o3)            [-, -, 4, z, ff23(x), a, b c , 0]
                        y  2

  Categories:Package combinatorics ·

Function: cyclep (c, n)

  Returns true if c is a valid cycle of order n namely, a list of non-repeated positive integers less or equal to n. Otherwise, it returns false.

  See also `permp`.

  Examples:

      (%i1) load("combinatorics")$

      (%i2) cyclep ([-2,3,4], 5);
      (%o2)                          false

      (%i3) cyclep ([2,3,4,2], 5);
      (%o3)                          false

      (%i4) cyclep ([6,3,4], 5);
      (%o4)                          false

      (%i5) cyclep ([6,3,4], 6);
      (%o5)                          true

  Categories:Package combinatorics ·

Function: perm_cycles (p)

  Returns permutation p as a product of cycles. The cycles are written in a canonical form, in which the lowest index in the cycle is placed in the first position.

  See also `perm_decomp`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_cycles ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                 [[1, 4], [2, 6, 5, 7]]

  Categories:Package combinatorics ·

Function: perm_decomp (p)

  Returns the minimum set of adjacent transpositions whose product equals the given permutation p.

  See also `perm_cycles`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_decomp ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2) [[6, 7], [5, 6], [6, 7], [3, 4], [4, 5], [2, 3], [3, 4],
                                  [4, 5], [5, 6], [1, 2], [2, 3], [3, 4]]

  Categories:Package combinatorics ·

Function: perm_inverse (p)

  Returns the inverse of a permutation of p, namely, a permutation q such that the products pq and qp are equal to the identity permutation: [1, 2, 3, …, n], where n is the length of p.

  See also `permult`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_inverse ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                [4, 7, 3, 1, 6, 2, 5, 8]

  Categories:Package combinatorics ·

Function: perm_length (p)

  Determines the minimum number of adjacent transpositions necessary to write permutation p as a product of adjacent transpositions. An adjacent transposition is a cycle with only two numbers, which are consecutive integers.

  See also `perm_decomp`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_length ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                           12

  Categories:Package combinatorics ·

Function: perm_lex_next (p)

  Returns the permutation that comes after the given permutation p, in the sequence of permutations in lexicographic order.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_lex_next ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

  Categories:Package combinatorics ·

Function: perm_lex_rank (p)

  Finds the position of permutation p, an integer from 1 to the degree n of the permutation, in the sequence of permutations in lexicographic order.

  See also `perm_lex_unrank` and `perms_lex`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_lex_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                          18255

  Categories:Package combinatorics ·

Function: perm_lex_unrank (n, i)

  Returns the n-degree permutation at position i (from 1 to n!) in the lexicographic ordering of permutations.

  See also `perm_lex_rank` and `perms_lex`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_lex_unrank (8, 18255);
      (%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

  Categories:Package combinatorics ·

Function: perm_next (p)

  Returns the permutation that comes after the given permutation p, in the sequence of permutations in Trotter-Johnson order.

  See also `perms`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_next ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

  Categories:Package combinatorics ·

Function: perm_parity (p)

  Finds the parity of permutation p: 0 if the minimum number of adjacent transpositions necessary to write permutation p as a product of adjacent transpositions is even, or 1 if that number is odd.

  See also `perm_decomp`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_parity ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                            0

  Categories:Package combinatorics ·

Function: perm_rank (p)

  Finds the position of permutation p, an integer from 1 to the degree n of the permutation, in the sequence of permutations in Trotter-Johnson order.

  See also `perm_unrank` and `perms`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
      (%o2)                          19729

  Categories:Package combinatorics ·

Function: perm_undecomp (cl, n)

  Converts the list of cycles cl of degree n into an n degree permutation, equal to their product.

  See also `perm_decomp`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_undecomp ([[1,6],[2,6,5,7]], 8);
      (%o2)                [5, 6, 3, 4, 7, 1, 2, 8]

  Categories:Package combinatorics ·

Function: perm_unrank (n, i)

  Returns the n-degree permutation at position i (from 1 to n!) in the Trotter-Johnson ordering of permutations.

  See also `perm_rank` and `perms`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) perm_unrank (8, 19729);
      (%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

  Categories:Package combinatorics ·

Function: permp (p)

  Returns true if p is a valid permutation namely, a list of length n, whose elements are all the positive integers from 1 to n, without repetitions. Otherwise, it returns false.

  Examples:

      (%i1) load("combinatorics")$

      (%i2) permp ([2,0,3,1]);
      (%o2)                          false

      (%i3) permp ([2,1,4,3]);
      (%o3)                          true

  Categories:Package combinatorics ·

Function: perms
`perms` (n)
`perms` (n, i)
`perms` (n, i, j)

  `perms(n)` returns a list of all n-degree permutations in the so-called Trotter-Johnson order.

  `perms(n, i)` returns the n-degree permutation which is at the ith position (from 1 to n!) in the Trotter-Johnson ordering of the permutations.

  `perms(n, i, j)` returns a list of the n-degree permutations between positions i and j in the Trotter-Johnson ordering of the permutations.

  The sequence of permutations in Trotter-Johnson order starts with the identity permutation and each consecutive permutation can be obtained from the previous one a by single adjacent transposition.

  See also `perm_next`, `perm_rank` and `perm_unrank`.

  Examples:

      (%i1) load("combinatorics")$

      (%i2) perms (4);
      (%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [4, 1, 2, 3],
      [4, 1, 3, 2], [1, 4, 3, 2], [1, 3, 4, 2], [1, 3, 2, 4],
      [3, 1, 2, 4], [3, 1, 4, 2], [3, 4, 1, 2], [4, 3, 1, 2],
      [4, 3, 2, 1], [3, 4, 2, 1], [3, 2, 4, 1], [3, 2, 1, 4],
      [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 3, 1], [4, 2, 3, 1],
      [4, 2, 1, 3], [2, 4, 1, 3], [2, 1, 4, 3], [2, 1, 3, 4]]

      (%i3) perms (4, 12);
      (%o3)                     [[4, 3, 1, 2]]

      (%i4) perms (4, 12, 14);
      (%o4)       [[4, 3, 1, 2], [4, 3, 2, 1], [3, 4, 2, 1]]

  Categories:Package combinatorics ·

Function: perms_lex
`perms_lex` (n)
`perms_lex` (n, i)
`perms_lex` (n, i, j)

  `perms_lex(n)` returns a list of all n-degree permutations in the so-called lexicographic order.

  `perms_lex(n, i)` returns the n-degree permutation which is at the ith position (from 1 to n!) in the lexicographic ordering of the permutations.

  `perms_lex(n, i, j)` returns a list of the n-degree permutations between positions i and j in the lexicographic ordering of the permutations.

  The sequence of permutations in lexicographic order starts with all the permutations with the lowest index, 1, followed by all permutations starting with the following index, 2, and so on. The permutations starting by an index i are the permutations of the first n integers different from i and they are also placed in lexicographic order, where the permutations with the lowest of those integers are placed first and so on.

  See also `perm_lex_next`, `perm_lex_rank` and `perm_lex_unrank`.

  Examples:

      (%i1) load("combinatorics")$

      (%i2) perms_lex (4);
      (%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2],
      [1, 4, 2, 3], [1, 4, 3, 2], [2, 1, 3, 4], [2, 1, 4, 3],
      [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1],
      [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1],
      [3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 2, 3], [4, 1, 3, 2],
      [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]

      (%i3) perms_lex (4, 12);
      (%o3)                     [[2, 4, 3, 1]]

      (%i4) perms_lex (4, 12, 14);
      (%o4)       [[2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2]]

  Categories:Package combinatorics ·

Function: permult (p_1, …, p_m)

  Returns the product of two or more permutations p_1, …, p_m.

  Example:

      (%i1) load("combinatorics")$

      (%i2) permult ([2,3,1], [3,1,2], [2,1,3]);
      (%o2)                        [2, 1, 3]

  Categories:Package combinatorics ·

Function: permute (p, l)

  Applies the permutation p to the elements of the list (or set) l.

  Example:

      (%i1) load("combinatorics")$

      (%i2) lis1: [a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                              2        x  1
      (%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                       y  2

      (%i3) permute ([4, 6, 3, 1, 7, 5, 2, 8], lis1);
                           1                 x     2
      (%o3)            [z, -, 4, a, ff23(x), -, b c , 0]
                           2                 y

  Categories:Package combinatorics ·

Function: random_perm (n)

  Returns a random permutation of degree n.

  See also `random_permutation`.

  Example:

      (%i1) load("combinatorics")$

      (%i2) random_perm (7);
      (%o2)                  [6, 3, 4, 7, 5, 1, 2]

  Categories:Package combinatorics ·