cp's OEIS Frontend

A362644 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of permutations of an n-set with k permutations.

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