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A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

%I A317449 #34 Dec 30 2020 14:57:14
%S A317449 1,2,2,3,6,3,5,21,16,5,7,52,72,32,7,11,141,306,216,65,11,15,327,1113,
%T A317449 1160,512,113,15,22,791,4033,6052,3737,1154,199,22,30,1780,13586,
%U A317449 28749,24325,10059,2317,323,30,42,4058,45514,133642,151994,82994,24854,4493,523,42
%N A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
%H A317449 Andrew Howroyd, <a href="/A317449/b317449.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%e A317449 The T(3,2) = 6 multiset partitions are {{1},{1,1}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}.
%e A317449 Triangle begins:
%e A317449     1
%e A317449     2    2
%e A317449     3    6    3
%e A317449     5   21   16    5
%e A317449     7   52   72   32    7
%e A317449    11  141  306  216   65   11
%e A317449    15  327 1113 1160  512  113   15
%e A317449    ...
%t A317449 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317449 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317449 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A317449 Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}]
%o A317449 (PARI)
%o A317449 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A317449 D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}
%o A317449 U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s}
%o A317449 M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))}
%o A317449 { my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ _Andrew Howroyd_, Dec 30 2020
%Y A317449 Row sums are A035310. First and last columns are both A000041.
%Y A317449 Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776.
%K A317449 nonn,tabl
%O A317449 1,2
%A A317449 _Gus Wiseman_, Aug 06 2018
%E A317449 Terms a(46) and beyond from _Andrew Howroyd_, Dec 30 2020