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A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)