A261096 A(i,j) = rank (in A055089) of the composition of the i-th and the j-th permutation in table A055089, which lists all finite permutations in reversed colexicographic ordering.

0, 1, 1, 2, 0, 2, 3, 4, 3, 3, 4, 5, 0, 2, 4, 5, 2, 1, 5, 5, 5, 6, 3, 5, 4, 1, 4, 6, 7, 7, 4, 0, 0, 3, 7, 7, 8, 6, 12, 1, 3, 2, 8, 6, 8, 9, 10, 13, 13, 2, 1, 9, 10, 9, 9, 10, 11, 14, 12, 18, 0, 10, 11, 6, 8, 10, 11, 8, 15, 16, 19, 19, 11, 8, 7, 11, 11, 11, 12, 9, 16, 17, 20, 18, 0, 9, 11, 10, 7, 10, 12, 13, 18, 17, 14, 21, 22, 1, 1, 10, 6, 6, 9, 13, 13, 14, 19, 6, 15, 22, 23, 2, 0, 14, 7, 9, 8, 14, 12, 14

Offset

0,4

Comments

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...

A(i,j) gives the rank (in ordering used by table A055089) of the permutation which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A055089 (note that the identity permutation is the 0th). Here the convention is that "permutations act of the left", thus, if p1 and p2 are permutations, then the product of p1 and p2 (p1 * p2) is defined such that (p1 * p2)(i) = p1(p2(i)) for i=1...

Each row and column is a permutation of A001477, because this is the Cayley table ("multiplication table") of an infinite enumerable group, namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of elements.

Links

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of array

Wikipedia, Cayley table

Formula

By conjugating with related permutations and arrays:

A(i,j) = A056019(A261097(A056019(i),A056019(j))).

A(i,j) = A060119(A261216(A060126(i),A060126(j))).

A(i,j) = A060120(A261217(A060127(i),A060127(j))).

Example

The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  4,  5,  2,  3,  7,  6, 10, 11,  8,  9, 18, ...
   2,  3,  0,  1,  5,  4, 12, 13, 14, 15, 16, 17,  6, ...
   3,  2,  5,  4,  0,  1, 13, 12, 16, 17, 14, 15, 19, ...
   4,  5,  1,  0,  3,  2, 18, 19, 20, 21, 22, 23,  7, ...
   5,  4,  3,  2,  1,  0, 19, 18, 22, 23, 20, 21, 13, ...
   6,  7,  8,  9, 10, 11,  0,  1,  2,  3,  4,  5, 14, ...
   7,  6, 10, 11,  8,  9,  1,  0,  4,  5,  2,  3, 20, ...
   8,  9,  6,  7, 11, 10, 14, 15, 12, 13, 17, 16,  0, ...
   9,  8, 11, 10,  6,  7, 15, 14, 17, 16, 12, 13, 21, ...
  10, 11,  7,  6,  9,  8, 20, 21, 18, 19, 23, 22,  1, ...
  11, 10,  9,  8,  7,  6, 21, 20, 23, 22, 18, 19, 15, ...
  12, 13, 14, 15, 16, 17,  2,  3,  0,  1,  5,  4,  8, ...
  ...
For A(1,2) (row=1, column=2, both starting from zero), we take as permutation p the permutation which has rank=1 in the ordering used by A055089, which is a simple transposition (1 2), which we can extend with fixed terms as far as we wish (e.g., like {2,1,3,4,5,...}), and as permutation q we take the permutation which has rank=2 (in the same list), which is {1,3,2}. We compose these from the left, so that the latter one, q, acts first, thus c(i) = p(q(i)), and the result is permutation {2,3,1}, which is listed as the 4th one in A055089, thus A(1,2) = 4.
For A(2,1) we compose those two permutations in opposite order, as d(i) = q(p(i)), which gives permutation {3,1,2} which is listed as the 3rd one in A055089, thus A(2,1) = 3.

Crossrefs

Transpose: A261097.

Row 0 & Column 0: A001477 (identity permutation).

Row 1: A261098.

Column 1: A004442.

Main diagonal: A261099.

Cf. tables A055089, A195663.

Cf. also A261216, A261217 (similar arrays, but using different orderings of permutations).

Permutations used in conjugation-formulas: A056019, A060119, A060120, A060126, A060127.

Sequence in context: A263624 A141693 A279679 * A257092 A374705 A028376

Adjacent sequences: A261093 A261094 A261095 * A261097 A261098 A261099

Keyword

nonn,tabl

Author

Antti Karttunen, Aug 26 2015

Status

approved