A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.
1, 1, 3, 13, 59, 313, 1847, 11977, 84483, 642405, 5228987, 45297249, 415582335, 4021374193, 40895428051, 435721370413, 4850551866619, 56282199807401, 679220819360775, 8508809310177481, 110454586096508563, 1483423600240661781, 20581786429087269819
Offset: 0
Keywords
Examples
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}}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..300
Crossrefs
Programs
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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 *)
Extensions
a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019