A326292 - OEIS

%I #8 Jun 28 2020 02:48:02

%S 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,

%T 3,5,7,11,15,22,30,43,57,80,105,142,186,248,320,421,539,698,889,1140,

%U 1438,1827,2291,2882,3593,4489,5559,6902,8503,10484,12853,15763

%N Number of crossing integer partitions of n.

%C 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.

%e The a(31) = 1 through a(36) = 7 partitions:

%e   21,10  21,10,1  21,10,2    21,10,3      21,10,4        21,10,5

%e                   21,10,1,1  21,10,2,1    21,10,2,2      21,10,3,2

%e                              21,10,1,1,1  21,10,3,1      21,10,4,1

%e                                           21,10,2,1,1    21,10,2,2,1

%e                                           21,10,1,1,1,1  21,10,3,1,1

%e                                                          21,10,2,1,1,1

%e                                                          21,10,1,1,1,1,1

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

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

%t Table[Length[Select[IntegerPartitions[n],croXQ[primeMS/@#]&]],{n,0,40}]

%Y The Heinz numbers of these partitions are given by A324170.

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

%K nonn

%O 0,34

%A _Gus Wiseman_, Oct 03 2019

%E More terms from _Jinyuan Wang_, Jun 28 2020