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Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.
10

%I #38 May 30 2023 16:12:05

%S 1,0,1,3,12,37,130,428,1481,5091,17979,64176,234311,869645,3295100,

%T 12720494,50083996,200964437,821845766,3423694821,14524845181,

%U 62725701708,275629610199,1231863834775,5597240308384,25844969339979,121224757935416,577359833539428,2791096628891679

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

%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="/A319077/b319077.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%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 R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^t))}

%o 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 Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

%Y Cf. A319748, A319755, A319778, A319781, A319790.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 27 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, May 30 2023