A317077 Number of connected multiset partitions of normal multisets of size n.
1, 1, 3, 8, 28, 110, 519, 2749, 16317, 106425, 755425, 5781956, 47384170, 413331955, 3818838624, 37213866876, 381108145231, 4088785729738, 45829237977692, 535340785268513, 6502943193997922, 81984445333355812, 1070848034863526547, 14467833457108560375, 201894571410270034773
Offset: 0
Keywords
Examples
The a(3) = 8 connected multiset partitions are (111), (1)(11), (1)(1)(1), (122), (2)(12), (112), (1)(12), (123).
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..500
Crossrefs
Programs
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Mathematica
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]]; allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]; Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,allnorm[n]}],{n,8}] -
PARI
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k]));u} seq(n)={my(u=vector(n, k, x*Ser(EulerT(vector(n,i,binomial(i+k-1,i)))))); Vec(1+vecsum(Connected(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*u[i])))))} \\ Andrew Howroyd, Jan 16 2023
Extensions
Terms a(9) and beyond from Andrew Howroyd, Jan 16 2023
Comments