A300383 In the ranked poset of integer partitions ordered by refinement, a(n) is the size of the lower ideal generated by the partition with Heinz number n.

1, 1, 2, 1, 3, 2, 5, 1, 3, 3, 7, 2, 11, 5, 5, 1, 15, 3, 22, 3, 8, 7, 30, 2, 6, 11, 4, 5, 42, 5, 56, 1, 11, 15, 11, 3, 77, 22, 17, 3, 101, 8, 135, 7, 7, 30, 176, 2, 14, 6, 23, 11, 231, 4, 15, 5, 33, 42, 297, 5, 385, 56, 11, 1, 23, 11, 490, 15, 45, 11, 627, 3

Offset: 1

Author

Gus Wiseman, Mar 04 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The size of the corresponding upper ideal is A317141(n). Chains are A213427(n) and maximal chains are A002846(n).

Examples

			The a(30) = 5 partitions are (321), (2211), (3111), (21111), (111111), with corresponding Heinz numbers: 30, 36, 40, 48, 64.
		

Links

• Robert Price, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273.

Cf. A317141, A317142, A317143.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[n]]]],{n,50}]

Formula

a(prime(n)) = A000041(n).

a(x * y) <= a(x) * a(y).