OFFSET
1,3
COMMENTS
Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms
EXAMPLE
From Gus Wiseman, Nov 09 2018: (Start)
Triangle begins:
1
1 2
1 2 3
1 4 4 5
1 4 8 7 7
1 6 12 16 12 11
1 6 17 25 28 19 15
1 8 22 43 49 48 30 22
1 8 30 58 87 88 77 45 30
...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
{{5}} {{1,4}} {{1,1,3}} {{1,1,1,2}} {{1,1,1,1,1}}
{{2,3}} {{1,2,2}} {{1},{1,1,2}} {{1},{1,1,1,1}}
{{1},{4}} {{1},{1,3}} {{1,1},{1,2}} {{1,1},{1,1,1}}
{{2},{3}} {{1},{2,2}} {{2},{1,1,1}} {{1},{1},{1,1,1}}
{{2},{1,2}} {{1},{1},{1,2}} {{1},{1,1},{1,1}}
{{3},{1,1}} {{1},{2},{1,1}} {{1},{1},{1},{1,1}}
{{1},{1},{3}} {{1},{1},{1},{2}} {{1},{1},{1},{1},{1}}
{{1},{2},{2}}
(End)
MAPLE
h:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Feb 17 2023
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]]]];
Table[Length[Join@@mps/@IntegerPartitions[n, {k}]], {n, 5}, {k, n}] (* Gus Wiseman, Nov 09 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, Jun 09 2000
STATUS
approved