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.

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, 84, 39, 11, 1, 1, 1, 64, 153, 331, 206, 92, 15, 1, 1, 1, 128, 428, 1300, 1087, 717, 170, 22, 1, 1, 1, 256, 1221, 5111, 5832, 5512, 1810, 360, 30, 1

Offset: 0

Author

Andrew Howroyd, May 09 2023

Keywords

• nonn
• tabl

Comments

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.

Examples

			Array begins:
=======================================================
n/k| 1  2   3    4     5       6        7         8 ...
---+---------------------------------------------------
0  | 1  1   1    1     1       1        1         1 ...
1  | 1  1   1    1     1       1        1         1 ...
2  | 1  2   4    8    16      32       64       128 ...
3  | 1  3   8   21    56     153      428      1221 ...
4  | 1  5  21   84   331    1300     5111     20144 ...
5  | 1  7  39  206  1087    5832    31949    178486 ...
6  | 1 11  92  717  5512   42601   333012   2635637 ...
7  | 1 15 170 1810 19252  208400  2303310  25936170 ...
8  | 1 22 360 5462 81937 1241302 19107225 299002252 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
• Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Crossrefs

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

Cf. A160870, A362827, A362903.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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)}
{ my(A=M(8)); for(n=1, #A~, print(A[n, ])) }

Formula

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

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

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

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