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

%I A255397 #14 Feb 04 2021 14:16:21
%S A255397 1,1,4,18,92,528,3356,23344,175984,1426520,12352600,113645488,
%T A255397 1105760224,11333738336,121957021744,1373618201360,16151326356192,
%U A255397 197796234588800,2517603785738752,33242912468993312,454583512625280256,6427749935432143072,93847133530055987840
%N A255397 Number of multimin-partitions of normal multisets of weight n.
%C A255397 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.
%H A255397 Andrew Howroyd, <a href="/A255397/b255397.txt">Table of n, a(n) for n = 0..200</a>
%H A255397 Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>
%e A255397 For a(3) = 18
%e A255397 [[1][2][3]]:[123]->[123]
%e A255397 [[1][23]]:[123]->[12]
%e A255397 [[13][2]]:[123]->[12]
%e A255397 [[12][3]]:[123]->[13]
%e A255397 [[123]]:[123]->[1]
%e A255397 [[1][2][2]]:[122]->[122]
%e A255397 [[1][22]]:[122]->[12]
%e A255397 [[12][2]]:[122]->[12]
%e A255397 [[122]]:[122]->[1]
%e A255397 [[1][1][2]]:[112]->[112]
%e A255397 [[1][12]]:[112]->[11]
%e A255397 [[12][1]]:[112]->[11]
%e A255397 [[11][2]]:[112]->[12]
%e A255397 [[112]]:[112]->[1]
%e A255397 [[1][1][1]]:[111]->[111]
%e A255397 [[1][11]]:[111]->[11]
%e A255397 [[11][1]]:[111]->[11]
%e A255397 [[111]]:[111]->[1]
%t A255397 mmcount[m_List] := mmcount[m] = If[Length[m] === 0, 0, 1 + Plus @@ mmcount /@ Union[Subsets[Rest[m]]]];
%t A255397 mmallnorm[n_Integer] := Function[s, Array[Count[s, y_ /; y <= #] + 1 &, n]] /@ Subsets[Range[n - 1] + 1];
%t A255397 Array[Plus @@ mmcount /@ mmallnorm[#] &, 13]
%o A255397 (PARI)
%o A255397 R(n,k)=Vec(prod(j=1, k, 1/(1 - x/(1-x + O(x^n))^j)) + O(x*x^n))
%o A255397 seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ _Andrew Howroyd_, Feb 04 2021
%Y A255397 Cf. A262671.
%K A255397 nonn
%O A255397 0,3
%A A255397 _Gus Wiseman_, Feb 22 2015
%E A255397 a(14)-a(15) from _Vaclav Kotesovec_, Feb 22 2015
%E A255397 a(0)=1 prepended and terms a(16) and beyond from _Andrew Howroyd_, Feb 04 2021