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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. 9
1, 1, 3, 13, 59, 313, 1847, 11977, 84483, 642405, 5228987, 45297249, 415582335, 4021374193, 40895428051, 435721370413, 4850551866619, 56282199807401, 679220819360775, 8508809310177481, 110454586096508563, 1483423600240661781, 20581786429087269819 (list; graph; refs; listen; history; text; internal format)
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}]

CROSSREFS

Cf. A001055, A007716, A045778, A255906, A281116, A317449, A317532, A317583, A317653, A317752, A317757, A317775.

Row sums of A327116.

Sequence in context: A151233 A151234 A330799 * A145942 A331517 A074437

Adjacent sequences:  A317773 A317774 A317775 * A317777 A317778 A317779

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 06 2018

EXTENSIONS

a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019

STATUS

approved

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Last modified July 12 23:28 EDT 2020. Contains 335669 sequences. (Running on oeis4.)