OFFSET
0,9
COMMENTS
Isomorphism is up to permutation of the elements of the n-set. Each permutation can be considered to be a set of disjoint directed cycles whose vertices cover the n-set. Permuting the elements of the n-set permutes each of the permutations in the multiset.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
FORMULA
T(0,k) = T(1,k) = 1.
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 3 4 5 6 7 ...
3 | 1 3 8 17 34 61 105 ...
4 | 1 5 28 159 888 4521 20916 ...
5 | 1 7 96 2655 76854 1882581 39122096 ...
6 | 1 11 495 88885 15719714 2271328951 274390124129 ...
7 | 1 15 2919 4255594 5341866647 5387750530872 4530149870111873 ...
...
PROG
(PARI)
B(n, k) = {n!*k^n}
K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
T(n, k) = {if(n==0, 1, my(s=0); forpart(q=n, s += permcount(q) * polcoef(exp(sum(m=1, k, K(R(q, m))*x^m/m, O(x*x^k))), k)); s/n!)}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 01 2023
STATUS
approved