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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 October 2 04:44 EDT 2023. Contains 365831 sequences. (Running on oeis4.)