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

0,3

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.

Links

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

Gus Wiseman, Comcategories and Multiorders

Example

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]

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]

Prog

(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

Crossrefs

Cf. A262671.

Sequence in context: A317135 A196150 A185298 * A337010 A081923 A020064

Adjacent sequences: A255394 A255395 A255396 * A255398 A255399 A255400

Keyword

nonn

Author

Gus Wiseman, Feb 22 2015

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

Status

approved