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
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