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A255397
Number of multimin-partitions of normal multisets of weight n.
17
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.
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
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