Andrew Howroyd, <a href="/A319077/b319077_1.txt">Table of n, a(n) for n = 0..50</a>

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Andrew Howroyd, <a href="/A319077/b319077_1.txt">Table of n, a(n) for n = 0..50</a>

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679

(PARI)

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

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}

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))}

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

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

nonn,more

nonn

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

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Cf. A319077, A319748, A319755, A319778, A319781, A319790.

Cf. A319077, A319748, A319755, A319778, A319781, A319790.

allocated for Gus WisemanNumber of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979

0,4

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

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

2: {{1},{2}}

3: {{1},{2,2}}

{{1},{2,3}}

{{1},{2},{3}}

4: {{1},{2,2,2}}

{{1},{2,3,3}}

{{1},{2,3,4}}

{{1,1},{2,2}}

{{1,2},{3,3}}

{{1,2},{3,4}}

{{1},{2},{1,2}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{2},{3,4}}

{{1},{3},{2,3}}

{{1},{2},{3},{4}}

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

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Gus Wiseman, Sep 27 2018

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