cp's OEIS Frontend

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

%I A319077 #38 May 30 2023 16:12:05
%S A319077 1,0,1,3,12,37,130,428,1481,5091,17979,64176,234311,869645,3295100,
%T A319077 12720494,50083996,200964437,821845766,3423694821,14524845181,
%U A319077 62725701708,275629610199,1231863834775,5597240308384,25844969339979,121224757935416,577359833539428,2791096628891679
%N A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.
%C A319077 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 A319077 Andrew Howroyd, <a href="/A319077/b319077.txt">Table of n, a(n) for n = 0..50</a>
%e A319077 Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:
%e A319077 2: {{1},{2}}
%e A319077 3: {{1},{2,2}}
%e A319077    {{1},{2,3}}
%e A319077    {{1},{2},{3}}
%e A319077 4: {{1},{2,2,2}}
%e A319077    {{1},{2,3,3}}
%e A319077    {{1},{2,3,4}}
%e A319077    {{1,1},{2,2}}
%e A319077    {{1,2},{3,3}}
%e A319077    {{1,2},{3,4}}
%e A319077    {{1},{2},{1,2}}
%e A319077    {{1},{2},{2,2}}
%e A319077    {{1},{2},{3,3}}
%e A319077    {{1},{2},{3,4}}
%e A319077    {{1},{3},{2,3}}
%e A319077    {{1},{2},{3},{4}}
%o A319077 (PARI)
%o A319077 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A319077 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 A319077 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 A319077 R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^t))}
%o A319077 a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t] - subst(x^(t*k)*u[t] + O(x*x^(n\2)), x, x^2), O(x*x^n) ))*if(k,1+x^k,1), n))) ); s/n!} \\ _Andrew Howroyd_, May 30 2023
%Y A319077 Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.
%Y A319077 Cf. A319748, A319755, A319778, A319781, A319790.
%K A319077 nonn
%O A319077 0,4
%A A319077 _Gus Wiseman_, Sep 27 2018
%E A319077 Terms a(11) and beyond from _Andrew Howroyd_, May 30 2023