login
A326292
Number of crossing integer partitions of n.
0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 80, 105, 142, 186, 248, 320, 421, 539, 698, 889, 1140, 1438, 1827, 2291, 2882, 3593, 4489, 5559, 6902, 8503, 10484, 12853, 15763
OFFSET
0,34
COMMENTS
A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. An integer partition is crossing if, by replacing each part with its multiset of prime indices, we obtain a crossing multiset partition.
EXAMPLE
The a(31) = 1 through a(36) = 7 partitions:
21,10 21,10,1 21,10,2 21,10,3 21,10,4 21,10,5
21,10,1,1 21,10,2,1 21,10,2,2 21,10,3,2
21,10,1,1,1 21,10,3,1 21,10,4,1
21,10,2,1,1 21,10,2,2,1
21,10,1,1,1,1 21,10,3,1,1
21,10,2,1,1,1
21,10,1,1,1,1,1
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
Table[Length[Select[IntegerPartitions[n], croXQ[primeMS/@#]&]], {n, 0, 40}]
CROSSREFS
The Heinz numbers of these partitions are given by A324170.
Sequence in context: A241728 A326589 A024794 * A195308 A218025 A091955
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 03 2019
EXTENSIONS
More terms from Jinyuan Wang, Jun 28 2020
STATUS
approved