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A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

%I A362826 #13 May 30 2023 14:10:38
%S A362826 1,1,1,1,1,1,1,1,2,1,1,1,4,3,1,1,1,8,8,5,1,1,1,16,21,21,7,1,1,1,32,56,
%T A362826 84,39,11,1,1,1,64,153,331,206,92,15,1,1,1,128,428,1300,1087,717,170,
%U A362826 22,1,1,1,256,1221,5111,5832,5512,1810,360,30,1
%N A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.
%C A362826 T(n,k) is also the number of nonisomorphic (k-1)-tuples of permutations of an n-set that pairwise commute. Isomorphism is up to permutation of the elements of the n-set.
%H A362826 Andrew Howroyd, <a href="/A362826/b362826.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A362826 Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv preprint arXiv:1304.2830 [math.CO], 2013.
%F A362826 Column k is the Euler transform of column k-1 of A160870.
%F A362826 T(n,k) = A362827(n,k) / n!.
%F A362826 G.f. of column k: exp(Sum_{i>=1} x^i*A160870(i,k)/i).
%F A362826 G.f. of column k > 1: 1/(Product_{i>=1} (1 - x^i)^A160870(i,k-1)).
%e A362826 Array begins:
%e A362826 =======================================================
%e A362826 n/k| 1  2   3    4     5       6        7         8 ...
%e A362826 ---+---------------------------------------------------
%e A362826 0  | 1  1   1    1     1       1        1         1 ...
%e A362826 1  | 1  1   1    1     1       1        1         1 ...
%e A362826 2  | 1  2   4    8    16      32       64       128 ...
%e A362826 3  | 1  3   8   21    56     153      428      1221 ...
%e A362826 4  | 1  5  21   84   331    1300     5111     20144 ...
%e A362826 5  | 1  7  39  206  1087    5832    31949    178486 ...
%e A362826 6  | 1 11  92  717  5512   42601   333012   2635637 ...
%e A362826 7  | 1 15 170 1810 19252  208400  2303310  25936170 ...
%e A362826 8  | 1 22 360 5462 81937 1241302 19107225 299002252 ...
%e A362826   ...
%o A362826 (PARI)
%o A362826 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A362826 M(n, m=n)={my(v=vector(m), u=vector(n, n, n==1)); for(j=1, #v, v[j]=concat([1], EulerT(u))~; u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
%o A362826 { my(A=M(8)); for(n=1, #A~, print(A[n, ])) }
%Y A362826 Columns k=1..4 are A000012, A000041, A061256, A226313.
%Y A362826 Cf. A160870, A362827, A362903.
%K A362826 nonn,tabl
%O A362826 0,9
%A A362826 _Andrew Howroyd_, May 09 2023