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%I #6 May 03 2023 21:36:40
%S 1,1,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,0,5,5,1,0,0,0,6,23,7,1,0,0,0,5,116,
%T 89,11,1,0,0,0,3,521,2494,484,15,1,0,0,0,1,1931,69366,87984,2904,22,1,
%U 0,0,0,0,5906,1592714,15456557,4250015,22002,30,1
%N Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-sets of permutations of an n-set.
%C 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 k-set.
%H Andrew Howroyd, <a href="/A362763/b362763.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F T(n,k) = 0 for k > n!.
%F T(n,k) = T(n, n!-k).
%e Array begins:
%e ====================================================================
%e n/k| 0 1 2 3 4 5 6 ...
%e ---+----------------------------------------------------------------
%e 0 | 1 1 0 0 0 0 0 ...
%e 1 | 1 1 0 0 0 0 0 ...
%e 2 | 1 2 1 0 0 0 0 ...
%e 3 | 1 3 5 6 5 3 1 ...
%e 4 | 1 5 23 116 521 1931 5906 ...
%e 5 | 1 7 89 2494 69366 1592714 30461471 ...
%e 6 | 1 11 484 87984 15456557 2209040882 263190866673 ...
%e 7 | 1 15 2904 4250015 5329123475 5366409944453 4503264576070573 ...
%e ...
%o (PARI)
%o B(n,k) = {n!*k^n}
%o K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
%o R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
%o 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}
%o T(n,k) = {if(n==0, k<=1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(R(q,m))*(x^m-x^(2*m))/m, O(x*x^k))), k)); s/n!)}
%Y Columns k=0..3 are A000012, A000041, A362764, A362765.
%Y Row sums are A362766.
%Y Cf. A362644.
%K nonn,tabl
%O 0,9
%A _Andrew Howroyd_, May 03 2023
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