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A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n. 1
1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} where x < z < y < t or z < x < t < y.

LINKS

Table of n, a(n) for n=1..63.

FORMULA

a(n) + A324326(n) = A318284(n).

EXAMPLE

The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:

  {{1,2,3,4}}

  {{1},{2,3,4}}

  {{2},{1,3,4}}

  {{3},{1,2,4}}

  {{4},{1,2,3}}

  {{1,2},{3,4}}

  {{1,4},{2,3}}

  {{1},{2},{3,4}}

  {{1},{3},{2,4}}

  {{1},{4},{2,3}}

  {{2},{3},{1,4}}

  {{2},{4},{1,3}}

  {{3},{1,2},{4}}

  {{1},{2},{3},{4}}

Missing from this list is {{1,3},{2,4}}.

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, Reverse[primeMS[n]]];

Table[Length[Select[facs[n], nonXQ[primeMS/@#]&]], {n, Array[Times@@Prime/@nrmptn[#]&, 30]}]

CROSSREFS

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.

Sequence in context: A114094 A179255 A332285 * A318284 A241321 A093936

Adjacent sequences:  A324322 A324323 A324324 * A324326 A324327 A324328

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 22 2019

STATUS

approved

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Last modified June 12 14:45 EDT 2021. Contains 344957 sequences. (Running on oeis4.)