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.

1, 2, 2, 3, 6, 3, 5, 21, 16, 5, 7, 52, 72, 32, 7, 11, 141, 306, 216, 65, 11, 15, 327, 1113, 1160, 512, 113, 15, 22, 791, 4033, 6052, 3737, 1154, 199, 22, 30, 1780, 13586, 28749, 24325, 10059, 2317, 323, 30, 42, 4058, 45514, 133642, 151994, 82994, 24854, 4493, 523, 42

Offset: 1

Author

Gus Wiseman, Aug 06 2018

Keywords

• nonn
• tabl

Examples

			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}}.
Triangle begins:
    1
    2    2
    3    6    3
    5   21   16    5
    7   52   72   32    7
   11  141  306  216   65   11
   15  327 1113 1160  512  113   15
   ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A035310. First and last columns are both A000041.

Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}]

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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]!)}
U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s}
M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))}
{ my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2020