A368099 - OEIS

%I #24 Jan 11 2024 20:03:57

%S 1,0,1,0,2,1,0,3,4,1,0,5,12,5,1,0,7,28,22,5,1,0,11,66,83,31,5,1,0,15,

%T 134,252,147,34,5,1,0,22,280,726,620,203,35,5,1,0,30,536,1946,2283,

%U 1069,235,35,5,1,0,42,1043,4982,7890,5019,1469,248,35,5,1

%N Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

%H Andrew Howroyd, <a href="/A368099/b368099.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%e Triangle begins:

%e     1

%e     0    1

%e     0    2    1

%e     0    3    4    1

%e     0    5   12    5    1

%e     0    7   28   22    5    1

%e     0   11   66   83   31    5    1

%e     0   15  134  252  147   34    5    1

%e     0   22  280  726  620  203   35    5    1

%e     0   30  536 1946 2283 1069  235   35    5    1

%e     0   42 1043 4982 7890 5019 1469  248   35    5    1

%e     ...

%e Row n = 4 counts the following representatives:

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

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

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

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

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

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

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

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

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

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

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

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

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];

%t brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];

%t Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

%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 G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}

%o T(n)={[Vecrev(p) | p <- Vec(G(n))]}

%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 11 2024

%Y Row sums are A316980, connected case A319557.

%Y For multiset partitions we have A317533, connected A322133.

%Y Counting connected components instead of edges gives A321194.

%Y For normal multiset partitions we have A330787, row sums A317776.

%Y For set multipartitions we have A334550.

%Y For set-systems we have A368096, row-sums A283877 (connected A300913).

%Y A000110 counts set-partitions, non-isomorphic A000041.

%Y A003465 counts covering set-systems, unlabeled A055621.

%Y A007716 counts non-isomorphic multiset partitions, connected A007718.

%Y A049311 counts non-isomorphic set multipartitions, connected A056156.

%Y A058891 counts set-systems, unlabeled A000612, connected A323818.

%Y Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Dec 31 2023