A317584 - OEIS

%I #8 Jan 01 2021 14:21:57

%S 1,4,6,19,14,113,30,584,1150,4023,112,119866,202,432061,5442765,

%T 16646712,594,738090160,980,13160013662,113864783987,39049423043,2510,

%U 44452496723053,19373518220009,21970704599961,8858890258339122,43233899006497146,9130,4019875470540832643

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

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

%F a(p) = 2*A000041(p) for prime p. - _Andrew Howroyd_, Jan 01 2021

%e The a(4) = 19 multiset partitions:

%e   {{1,1,1,1}}, {{1,1},{1,1}}, {{1},{1},{1},{1}},

%e   {{1,1,1,2}}, {{1,1},{1,2}}, {{1},{1},{1},{2}},

%e   {{1,1,2,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1},{1},{2},{2}},

%e   {{1,1,2,3}}, {{1,1},{2,3}}, {{1,2},{1,3}}, {{1},{1},{2},{3}},

%e   {{1,2,3,4}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}, {{1},{2},{3},{4}}.

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t Table[Length[Select[Join@@mps/@strnorm[n],SameQ@@Length/@#&]],{n,6}]

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o cycleIndex(n)={sum(n=1, n, x^n*sumdiv(n, d, sApplyCI(symGroupCycleIndex(d), d, symGroupCycleIndex(n/d), n/d))) + O(x*x^n)}

%o StronglyNormalLabelingsSeq(cycleIndex(15)) \\ _Andrew Howroyd_, Jan 01 2021

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 01 2018

%E Terms a(9) and beyond from _Andrew Howroyd_, Jan 01 2021