|
%I #9 May 02 2023 17:49:24
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,4,8,5,1,1,1,5,17,28,7,1,1,1,6,34,
%T 159,96,11,1,1,1,7,61,888,2655,495,15,1,1,1,8,105,4521,76854,88885,
%U 2919,22,1,1,1,9,170,20916,1882581,15719714,4255594,22024,30,1
%N Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of permutations of an n-set with k permutations.
%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 multiset.
%H Andrew Howroyd, <a href="/A362644/b362644.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F T(0,k) = T(1,k) = 1.
%e Array begins:
%e ====================================================================
%e n/k| 0 1 2 3 4 5 6 ...
%e ---+----------------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 1 2 3 4 5 6 7 ...
%e 3 | 1 3 8 17 34 61 105 ...
%e 4 | 1 5 28 159 888 4521 20916 ...
%e 5 | 1 7 96 2655 76854 1882581 39122096 ...
%e 6 | 1 11 495 88885 15719714 2271328951 274390124129 ...
%e 7 | 1 15 2919 4255594 5341866647 5387750530872 4530149870111873 ...
%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, 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!)}
%Y Columns k=0..3 are A000012, A000041, A362645, A362646.
%Y Rows n=3 is A002626.
%Y Main diagonal is A362647.
%Y Cf. A362648.
%K nonn,tabl
%O 0,9
%A _Andrew Howroyd_, May 01 2023
|
