%I #13 May 30 2023 14:10:38

%S 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 84,39,11,1,1,1,64,153,331,206,92,15,1,1,1,128,428,1300,1087,717,170,

%U 22,1,1,1,256,1221,5111,5832,5512,1810,360,30,1

%N 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 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 Andrew Howroyd, <a href="/A362826/b362826.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).

%H Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

%F Column k is the Euler transform of column k-1 of A160870.

%F T(n,k) = A362827(n,k) / n!.

%F G.f. of column k: exp(Sum_{i>=1} x^i*A160870(i,k)/i).

%F G.f. of column k > 1: 1/(Product_{i>=1} (1 - x^i)^A160870(i,k-1)).

%e Array begins:

%e =======================================================

%e n/k| 1 2 3 4 5 6 7 8 ...

%e ---+---------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 ...

%e 1 | 1 1 1 1 1 1 1 1 ...

%e 2 | 1 2 4 8 16 32 64 128 ...

%e 3 | 1 3 8 21 56 153 428 1221 ...

%e 4 | 1 5 21 84 331 1300 5111 20144 ...

%e 5 | 1 7 39 206 1087 5832 31949 178486 ...

%e 6 | 1 11 92 717 5512 42601 333012 2635637 ...

%e 7 | 1 15 170 1810 19252 208400 2303310 25936170 ...

%e 8 | 1 22 360 5462 81937 1241302 19107225 299002252 ...

%e ...

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 { my(A=M(8)); for(n=1, #A~, print(A[n, ])) }

%Y Columns k=1..4 are A000012, A000041, A061256, A226313.

%Y Cf. A160870, A362827, A362903.

%K nonn,tabl

%O 0,9

%A _Andrew Howroyd_, May 09 2023