A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 1, 8, 18, 24, 1, 1, 1, 16, 48, 120, 120, 1, 1, 1, 32, 126, 504, 840, 720, 1, 1, 1, 64, 336, 2016, 4680, 7920, 5040, 1, 1, 1, 128, 918, 7944, 24720, 66240, 75600, 40320, 1, 1, 1, 256, 2568, 31200, 130440, 516240, 856800, 887040, 362880, 1

Offset

0,9

Comments

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

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.

Formula

T(n,k) = n!*A362826(n,k) for k > 0.

Example

Array begins:
========================================================
n/k| 0    1     2      3       4        5          6 ...
---+----------------------------------------------------
0  | 1    1     1      1       1        1          1 ...
1  | 1    1     1      1       1        1          1 ...
2  | 1    2     4      8      16       32         64 ...
3  | 1    6    18     48     126      336        918 ...
4  | 1   24   120    504    2016     7944      31200 ...
5  | 1  120   840   4680   24720   130440     699840 ...
6  | 1  720  7920  66240  516240  3968640   30672720 ...
7  | 1 5040 75600 856800 9122400 97030080 1050336000 ...
  ...

Prog

(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
M(n, m=n)={my(v=vector(m+1), u=vector(n, n, n==1), f=vector(n, n, n!)); v[1]=vectorv(n+1, i, 1); for(j=1, #v-1, my(t=EulerT(u)); v[j+1]=vectorv(n+1, i, i--; if(i, f[i]*t[i], 1)); u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(7)); for(n=1, #A, print(A[n, ])) }

Crossrefs

Columns k=0..3 are A000012, A000142, A053529, A072169.

Main diagonal is A362828.

Cf. A362824, A362826.

Sequence in context: A140274 A095231 A303697 * A342413 A202019 A295685

Adjacent sequences: A362824 A362825 A362826 * A362828 A362829 A362830

Keyword

nonn,tabl

Author

Andrew Howroyd, May 08 2023

Status

approved