A255397 Number of multimin-partitions of normal multisets of weight n.

1, 1, 4, 18, 92, 528, 3356, 23344, 175984, 1426520, 12352600, 113645488, 1105760224, 11333738336, 121957021744, 1373618201360, 16151326356192, 197796234588800, 2517603785738752, 33242912468993312, 454583512625280256, 6427749935432143072, 93847133530055987840

Offset: 0

Author

Gus Wiseman, Feb 22 2015

Keywords

• nonn

Comments

A multiset is normal if its entries span an initial interval of positive integers. A multimin-partition is any sequence of multisets whose minima are weakly increasing. In a suitable category (see example) multimin-partitions m=(m_1,...,m_k) are morphisms m : U(m_1,...,m_k) -> {min(m_1),...,min(m_k)} where U denotes multiset union and min denotes minimum.

Examples

			For a(3) = 18
[[1][2][3]]:[123]->[123]
[[1][23]]:[123]->[12]
[[13][2]]:[123]->[12]
[[12][3]]:[123]->[13]
[[123]]:[123]->[1]
[[1][2][2]]:[122]->[122]
[[1][22]]:[122]->[12]
[[12][2]]:[122]->[12]
[[122]]:[122]->[1]
[[1][1][2]]:[112]->[112]
[[1][12]]:[112]->[11]
[[12][1]]:[112]->[11]
[[11][2]]:[112]->[12]
[[112]]:[112]->[1]
[[1][1][1]]:[111]->[111]
[[1][11]]:[111]->[11]
[[11][1]]:[111]->[11]
[[111]]:[111]->[1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Gus Wiseman, Comcategories and Multiorders

Crossrefs

Cf. A262671.

Programs

Mathematica

mmcount[m_List] := mmcount[m] = If[Length[m] === 0, 0, 1 + Plus @@ mmcount /@ Union[Subsets[Rest[m]]]];
mmallnorm[n_Integer] := Function[s, Array[Count[s, y_ /; y <= #] + 1 &, n]] /@ Subsets[Range[n - 1] + 1];
Array[Plus @@ mmcount /@ mmallnorm[#] &, 13]

PARI

R(n,k)=Vec(prod(j=1, k, 1/(1 - x/(1-x + O(x^n))^j)) + O(x*x^n))
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 04 2021

Extensions

a(14)-a(15) from Vaclav Kotesovec, Feb 22 2015

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Feb 04 2021