

A318560


Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.


3



1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25
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OFFSET

1,3


COMMENTS

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.
The prime indices of n are the nth row of A296150.


LINKS

Table of n, a(n) for n=1..40.


EXAMPLE

The a(18) = 17 combinatory separations of {1,1,2,2,3}:
{11223}
{1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
{1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
{1,1,1,11} {1,1,1,12}
{1,1,1,1,1}


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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]], i}, {i, Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize, mps[nrmptn[n]], {2}]]], {n, 20}]


CROSSREFS

Cf. A007716, A056239, A255906, A269134, A296150, A305936.
Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.
Sequence in context: A120887 A155860 A318285 * A030385 A253553 A031250
Adjacent sequences: A318557 A318558 A318559 * A318561 A318562 A318563


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Aug 28 2018


STATUS

approved



