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, 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.

Links

Table of n, a(n) for n=0..65.

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.

Cf. A000108, A002662, A016098, A056239, A112798, A324172, A324326, A326210, A326252, A326332.

Sequence in context: A241728 A326589 A024794 * A195308 A218025 A091955

Adjacent sequences: A326289 A326290 A326291 * A326293 A326294 A326295

Keyword

nonn

Author

Gus Wiseman, Oct 03 2019

Extensions

More terms from Jinyuan Wang, Jun 28 2020

Status

approved