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A261216

A(i,j) = rank (in A060117) of the composition of the i-th and the j-th permutation in table A060117, which lists all finite permutations.
(history; published version)

#26 by N. J. A. Sloane at Thu Sep 24 01:40:50 EDT 2015

STATUS

proposed

approved

#25 by Antti Karttunen at Tue Sep 15 13:38:43 EDT 2015

STATUS

editing

proposed

#24 by Antti Karttunen at Tue Sep 15 13:34:34 EDT 2015

COMMENTS

Because Each row and column is a permutation of A001477, because this is a the Cayley table ("multiplication table" (Cayley table) of an infinite enumerable group, each row and column is a permutation namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of A001477elements.

#23 by Antti Karttunen at Sun Aug 30 12:33:46 EDT 2015

EXAMPLE

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 A060117, thus A(1,2,1) = 3.

Discussion

Mon Sep 14

09:12

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#22 by Antti Karttunen at Sun Aug 30 11:45:50 EDT 2015

CROSSREFS

Cf. also A060117, A060118, A261096, A261097.

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

#21 by Antti Karttunen at Sun Aug 30 11:37:38 EDT 2015

FORMULA

A(i,j) = A060127(A261097(A060120(i),A060120(j))).

#20 by Antti Karttunen at Sun Aug 30 10:46:43 EDT 2015

FORMULA

By conjugating with related permutations and arrays:

A(i,j) = A060125(A261217(A060125(i),A060125(j))).

A(i,j) = A060126(A261096(A060119(i),A060119(j))).

#19 by Antti Karttunen at Sun Aug 30 10:05:05 EDT 2015

NAME

A(r,ci,j) = rank (in A060117) of the composition of A060117(r) the i-th and A060117(c) the j-th permutation in table A060117, which lists all finite permutations.

#18 by Antti Karttunen at Sun Aug 30 09:24:19 EDT 2015

COMMENTS

A(i,j) gives the rank of the permutation (in ordering used by table A060117) which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060117 (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...

EXAMPLE

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 A060117, 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 5th one in A060117 (the identity permutation is the 0th), , thus A(1,2) = 5.

#17 by Antti Karttunen at Sun Aug 30 09:17:38 EDT 2015

COMMENTS

A(i,j) gives the rank of a the permutation (in ordering used by table A060117) which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060117. 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 A060118 of the composition of the i-th and the j-th permutation in A060118, when convention is that "permutations act on the right".