A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 11, 4, 1, 1, 16, 43, 24, 5, 1, 1, 32, 171, 176, 46, 6, 1, 1, 64, 683, 1376, 611, 80, 7, 1, 1, 128, 2731, 10944, 9281, 1864, 130, 8, 1, 1, 256, 10923, 87424, 146445, 54384, 5161, 200, 9, 1, 1, 512, 43691, 699136, 2334181, 1696352, 285939, 13184, 295, 10, 1

Offset: 0

Author

Andrew Howroyd, May 11 2023

Keywords

• nonn
• tabl

Comments

Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

Examples

			Array begins:
======================================================
n/k| 0 1   2     3       4          5            6 ...
---+--------------------------------------------------
0  | 1 1   1     1       1          1            1 ...
1  | 1 2   4     8      16         32           64 ...
2  | 1 3  11    43     171        683         2731 ...
3  | 1 4  24   176    1376      10944        87424 ...
4  | 1 5  46   611    9281     146445      2334181 ...
5  | 1 6  80  1864   54384    1696352     53885632 ...
6  | 1 7 130  5161  285939   17562679   1110290303 ...
7  | 1 8 200 13184 1372224  165343616  20774749952 ...
8  | 1 9 295 31532 6101080 1436647664 358238974304 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Crossrefs

Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.

Rows n=1..3 are A000079, A007583, A103334(n+1).

Cf. A022166, A362648, A362824, A362826.

Programs

PARI

\\ B(n, k) is A022166.
B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
C(k,n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k,j+1), 1 - x + O(x*x^n)))
M(n,m=n) = Mat(vector(m+1, k, C(k-1, n)~))
{ my(A=M(7)); for(i=1, #A, print(A[i,])) }

Formula

G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).