|
|
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.
|
|
6
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
EXAMPLE
|
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
...
|
|
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}]
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|