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A261217 A(i,j) = rank (in A060118) of the composition of the i-th and the j-th permutation in table A060118, which lists all finite permutations. 6
0, 1, 1, 2, 0, 2, 3, 3, 5, 3, 4, 2, 0, 4, 4, 5, 5, 4, 1, 3, 5, 6, 4, 3, 5, 5, 2, 6, 7, 7, 1, 2, 1, 4, 7, 7, 8, 6, 8, 0, 0, 0, 14, 6, 8, 9, 9, 11, 9, 2, 1, 15, 15, 11, 9, 10, 8, 6, 10, 10, 3, 22, 14, 12, 10, 10, 11, 11, 10, 7, 9, 11, 23, 23, 16, 13, 9, 11, 12, 10, 9, 11, 11, 8, 0, 22, 21, 17, 17, 8, 12, 13, 13, 7, 8, 7, 10, 1, 1, 19, 20, 13, 16, 19, 13, 14, 12, 14, 6, 6, 6, 12, 0, 2, 18, 18, 12, 8, 18, 14 (list; table; graph; refs; listen; history; text; internal format)
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 A060118) 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 A060118 (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...

Equally, A(i,j) gives the rank in A060117 of the composition of the i-th and the j-th permutation in A060117, when convention is that "permutations act on the right".

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..7259; the first 120 antidiagonals of array

Wikipedia, Cayley table

FORMULA

By conjugating with related permutations and arrays:

A(i,j) = A060125(A261216(A060125(i),A060125(j))).

A(i,j) = A060127(A261096(A060120(i),A060120(j))).

A(i,j) = A060126(A261097(A060119(i),A060119(j))).

EXAMPLE

The top left corner of the array:

   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...

   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, ...

   2,  5,  0,  4,  3,  1,  8, 11,  6, 10,  9,  7, 14, ...

   3,  4,  1,  5,  2,  0,  9, 10,  7, 11,  8,  6, 15, ...

   4,  3,  5,  1,  0,  2, 10,  9, 11,  7,  6,  8, 16, ...

   5,  2,  4,  0,  1,  3, 11,  8, 10,  6,  7,  9, 17, ...

   6,  7, 14, 15, 22, 23,  0,  1, 12, 13, 18, 19,  8, ...

   7,  6, 15, 14, 23, 22,  1,  0, 13, 12, 19, 18,  9, ...

   8, 11, 12, 16, 21, 19,  2,  5, 14, 17, 20, 23,  6, ...

   9, 10, 13, 17, 20, 18,  3,  4, 15, 16, 21, 22,  7, ...

  10,  9, 17, 13, 18, 20,  4,  3, 16, 15, 22, 21, 11, ...

  11,  8, 16, 12, 19, 21,  5,  2, 17, 14, 23, 20, 10, ...

  12, 19,  8, 21, 16, 11, 14, 23,  2, 20, 17,  5,  0, ...

  ...

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 A060118, 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 3rd one in A060118, thus A(1,2) = 3.

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 5th one in A060118, thus A(2,1) = 5.

CROSSREFS

Transpose: A261216.

Row 0 & Column 0: A001477 (identity permutation)

Row 1: A004442.

Column 1: A261218.

Main diagonal: A261219.

Cf. also A060117, A060118, A261096, A261097.

Cf. also A089839.

Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.

Sequence in context: A003987 A141692 A261097 * A245230 A284268 A063180

Adjacent sequences:  A261214 A261215 A261216 * A261218 A261219 A261220

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Aug 26 2015

STATUS

approved

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Last modified February 23 13:25 EST 2018. Contains 299581 sequences. (Running on oeis4.)