A362903 -id:A362903 - cp's OEIS frontend

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

Andrew Howroyd, May 09 2023

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.

			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 ...
  ...

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.

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

Cf. A160870, A362827, A362903.

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, ])) }

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)).

A362904 Number of nonisomorphic ordered triples of involutions on a (2n)-set that pairwise commute.

Original entry on oeis.org

1, 8, 43, 176, 611, 1864, 5161, 13184, 31532, 71264, 153444, 316608, 629236, 1209312, 2255324, 4093056, 7246690, 12542736, 21262582, 35359456, 57767766, 92832784, 146908290, 229169792, 352721676, 536076640, 805132548, 1195771840, 1757278132

Offset: 0

Andrew Howroyd, May 11 2023

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8, -21, 0, 106, -176, -70, 512, -435, -392, 1001, -512, -580, 1120, -580, -512, 1001, -392, -435, 512, -70, -176, 106, 0, -21, 8, -1).

Column k=3 of A362903.

PARI
```
C(3,30) \\ C(k,n) defined in A362903.
```

G.f.: 1/((1 - x)^16*(1 + x)^8*(1 + x^2)).

cp's OEIS Frontend

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.

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