A321407 - OEIS

%I #8 Jan 17 2023 18:28:29

%S 1,0,1,2,7,13,47,111,367,1057,3474,11116,38106,131235,470882,1720959,

%T 6472129,24860957,97779665,392642763,1610045000,6732768139,

%U 28699327441,124600601174,550684155992,2476019025827,11320106871951,52598300581495,248265707440448,1189855827112636,5787965846277749

%N Number of non-isomorphic multiset partitions of weight n with no constant parts.

%C Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which every row has at least two nonzero entries.

%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%H Andrew Howroyd, <a href="/A321407/b321407.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions:

%e   {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}

%e            {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}

%e                       {{1,2,3,3}}    {{1,2,2,3,3}}

%e                       {{1,2,3,4}}    {{1,2,3,3,3}}

%e                       {{1,2},{1,2}}  {{1,2,3,4,4}}

%e                       {{1,2},{3,4}}  {{1,2,3,4,5}}

%e                       {{1,3},{2,3}}  {{1,2},{1,2,2}}

%e                                      {{1,2},{2,3,3}}

%e                                      {{1,2},{3,4,4}}

%e                                      {{1,2},{3,4,5}}

%e                                      {{1,3},{2,3,3}}

%e                                      {{1,4},{2,3,4}}

%e                                      {{2,3},{1,2,3}}

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%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 K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}

%o S(q, t, k)={sum(j=1, #q, if(t%q[j]==0, q[j]))*vector(k,i,1)}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(x*Ser(K(q, t, n\t)-S(q, t, n\t))/t, x, x^t) )), n)); s/n!)} \\ _Andrew Howroyd_, Jan 17 2023

%Y Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320798, A320801, A320808, A321760.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 29 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 17 2023