A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 1, 8, 10, 10, 1, 1, 1, 16, 22, 52, 26, 1, 1, 1, 32, 46, 232, 196, 76, 1, 1, 1, 64, 94, 976, 1016, 1216, 232, 1, 1, 1, 128, 190, 4000, 4576, 12496, 5944, 764, 1, 1, 1, 256, 382, 16192, 19376, 111376, 73648, 42400, 2620, 1

Offset: 0

Author

Andrew Howroyd, May 06 2023

Keywords

• nonn
• tabl

Comments

Two involutions x,y on [n] commute if x*y = y*x.

Examples

			Array begins:
===========================================================
n/k| 0   1    2     3      4       5        6         7 ...
---+-------------------------------------------------------
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   4   10    22     46      94      190       382 ...
4  | 1  10   52   232    976    4000    16192     65152 ...
5  | 1  26  196  1016   4576   19376    79696    323216 ...
6  | 1  76 1216 12496 111376  936976  7680016  62177296 ...
7  | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
  ...
		

Links

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

Crossrefs

Columns k=0..3 are A000012, A000085, A362819, A362825.

Rows n=2..3 are A000079, A033484.

Main diagonal is A362823.

Cf. A022166, A362648.

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)}
T(n,k)={if(n==0, 1, n!*polcoef(exp(sum(j=0, min(k,logint(n,2)), B(k,j)*x^(2^j)/2^j, O(x*x^n))), n))}

Formula

T(0,k) = T(1,k) = 1.