A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

1, 4, 6, 19, 14, 113, 30, 584, 1150, 4023, 112, 119866, 202, 432061, 5442765, 16646712, 594, 738090160, 980, 13160013662, 113864783987, 39049423043, 2510, 44452496723053, 19373518220009, 21970704599961, 8858890258339122, 43233899006497146, 9130, 4019875470540832643

Offset

1,2

Comments

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Links

Table of n, a(n) for n=1..30.

Formula

a(p) = 2*A000041(p) for prime p. - Andrew Howroyd, Jan 01 2021

Example

The a(4) = 19 multiset partitions:
  {{1,1,1,1}}, {{1,1},{1,1}}, {{1},{1},{1},{1}},
  {{1,1,1,2}}, {{1,1},{1,2}}, {{1},{1},{1},{2}},
  {{1,1,2,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1},{1},{2},{2}},
  {{1,1,2,3}}, {{1,1},{2,3}}, {{1,2},{1,3}}, {{1},{1},{2},{3}},
  {{1,2,3,4}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}, {{1},{2},{3},{4}}.

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], SameQ@@Length/@#&]], {n, 6}]

Prog

(PARI) \\ See links in A339645 for combinatorial species functions.
cycleIndex(n)={sum(n=1, n, x^n*sumdiv(n, d, sApplyCI(symGroupCycleIndex(d), d, symGroupCycleIndex(n/d), n/d))) + O(x*x^n)}
StronglyNormalLabelingsSeq(cycleIndex(15)) \\ Andrew Howroyd, Jan 01 2021

Crossrefs

Cf. A000005, A000041, A007716, A038041, A255906, A298422, A306017, A306018, A306019, A306020, A306021, A317583.

Sequence in context: A057393 A196570 A274992 * A012928 A013160 A153777

Adjacent sequences: A317581 A317582 A317583 * A317585 A317586 A317587

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2021

Status

approved