A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

1, 1, 2, 8, 70, 1012, 21944, 665708, 26917492, 1399033348, 90878863352, 7214384973908, 687197223963640, 77354805301801012, 10158257981179981304, 1539156284259756811748, 266517060496258245459352, 52301515332984084095078308, 11546416513975694879642736152

Offset

0,3

Comments

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

A finite multiset is normal if it covers an initial interval of positive integers.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Example

The a(0) = 1 through a(3) = 8 multisystems:
  {}  {1}  {1,1}  {{1},{1,1}}
           {1,2}  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

Mathematica

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1<Length[#]<Length[m]&]}], m];
Table[Sum[Length[Select[totm[m], Depth[#]==If[n<=1, 2, n]&]], {m, allnorm[n]}], {n, 5}]

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

Row sums of A330778.

The case with all atoms equal is A000111.

The case with all atoms different is A006472.

The version allowing all depths is A330655.

The unlabeled version is A330663.

The version where the atoms are the prime indices of n is A330665.

The strongly normal version is A330675.

The version where the degrees are the prime indices of n is A330728.

Multiset partitions of normal multisets are A255906.

Series-reduced rooted trees with normal leaves are A316651.

Cf. A000669, A001055, A005121, A005804, A318812, A330469, A330474, A330654, A330664, A330677, A330679.

Sequence in context: A239843 A099729 A123117 * A062733 A180687 A356811

Adjacent sequences: A330673 A330674 A330675 * A330677 A330678 A330679

Keyword

nonn

Author

Gus Wiseman, Dec 30 2019

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2019

Status

approved