This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A261216 #26 Sep 24 2015 01:40:50
%S A261216 0,1,1,2,0,2,3,5,3,3,4,4,0,2,4,5,3,1,4,5,5,6,2,5,5,3,4,6,7,7,4,1,2,1,
%T A261216 7,7,8,6,14,0,0,0,8,6,8,9,11,15,15,1,2,9,11,9,9,10,10,12,14,22,3,10,
%U A261216 10,6,8,10,11,9,13,16,23,23,11,9,7,10,11,11,12,8,17,17,21,22,0,8,11,11,9,10,12,13,19,16,13,20,19,1,1,10,7,8,7,13,13,14,18,8,12,18,18,2,0,12,6,6,6,14,12,14
%N 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.
%C A261216 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), ...
%C A261216 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...
%C A261216 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".
%C A261216 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.
%H A261216 Antti Karttunen, <a href="/A261216/b261216.txt">Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of array</a>
%H A261216 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cayley_table">Cayley table</a>
%F A261216 By conjugating with related permutations and arrays:
%F A261216 A(i,j) = A060125(A261217(A060125(i),A060125(j))).
%F A261216 A(i,j) = A060126(A261096(A060119(i),A060119(j))).
%F A261216 A(i,j) = A060127(A261097(A060120(i),A060120(j))).
%e A261216 The top left corner of the array:
%e A261216 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
%e A261216 1, 0, 5, 4, 3, 2, 7, 6, 11, 10, 9, 8, 19, ...
%e A261216 2, 3, 0, 1, 5, 4, 14, 15, 12, 13, 17, 16, 8, ...
%e A261216 3, 2, 4, 5, 1, 0, 15, 14, 16, 17, 13, 12, 21, ...
%e A261216 4, 5, 3, 2, 0, 1, 22, 23, 21, 20, 18, 19, 16, ...
%e A261216 5, 4, 1, 0, 2, 3, 23, 22, 19, 18, 20, 21, 11, ...
%e A261216 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 14, ...
%e A261216 7, 6, 11, 10, 9, 8, 1, 0, 5, 4, 3, 2, 23, ...
%e A261216 8, 9, 6, 7, 11, 10, 12, 13, 14, 15, 16, 17, 2, ...
%e A261216 9, 8, 10, 11, 7, 6, 13, 12, 17, 16, 15, 14, 20, ...
%e A261216 10, 11, 9, 8, 6, 7, 18, 19, 20, 21, 22, 23, 17, ...
%e A261216 11, 10, 7, 6, 8, 9, 19, 18, 23, 22, 21, 20, 5, ...
%e A261216 12, 13, 14, 15, 16, 17, 8, 9, 6, 7, 11, 10, 0, ...
%e A261216 ...
%e A261216 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, thus A(1,2) = 5.
%e A261216 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(2,1) = 3.
%Y A261216 Transpose: A261217.
%Y A261216 Row 0 & Column 0: A001477 (identity permutation).
%Y A261216 Row 1: A261218.
%Y A261216 Column 1: A004442.
%Y A261216 Main diagonal: A261219.
%Y A261216 Cf. also A060117, A060118, A261096, A261097.
%Y A261216 Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.
%K A261216 nonn,tabl
%O A261216 0,4
%A A261216 _Antti Karttunen_, Aug 26 2015