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Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.
2

%I #12 Jan 16 2024 17:33:00

%S 1,0,1,1,4,4,16,22,70,132,375,848,2428,6256,18333,52560,161436,500887,

%T 1624969,5384625,18438815,64674095,233062429,859831186,3248411250,

%U 12545820860,49508089411,199410275018,819269777688,3430680180687,14633035575435,63535672197070

%N Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.

%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="/A321677/b321677.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(2) = 1 through a(6) = 16 set multipartitions:

%e {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}} {{1,2,3,4,5,6}}

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

%e {{1,2},{3,4}} {{1,4},{2,3,4}} {{1,2},{3,4,5,6}}

%e {{1,3},{2,3}} {{2,3},{1,2,3}} {{1,2,3},{4,5,6}}

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

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

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

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

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

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

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

%e {{1,2},{3,4},{5,6}}

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

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

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

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

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g), n)); s/n!)} \\ _Andrew Howroyd_, Jan 16 2024

%Y Cf. A000219, A007716, A049311, A283877, A302545, A316983, A319616.

%Y Cf. A320797, A320798, A320804, A320811, A320812, A321404, A321406.

%K nonn

%O 0,5

%A _Gus Wiseman_, Nov 16 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019