OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..300
EXAMPLE
The a(3) = 13 strict multiset partitions:
{{1,1,1}}, {{1},{1,1}},
{{1,2,2}}, {{1},{2,2}}, {{2},{1,2}},
{{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
{{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.
MAPLE
C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23); # Alois P. Heinz, Sep 16 2019
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]]]];
allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n], UnsameQ@@#&]], {n, 9}]
(* Second program: *)
c := Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {k, 0, n}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 06 2018
EXTENSIONS
a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019
STATUS
approved