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

1,3

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).

Links

Table of n, a(n) for n=1..72.

Formula

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

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

Example

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

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}]

Crossrefs

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

Cf. A317141, A317142, A317143.

Sequence in context: A182167 A117364 A193615 * A120250 A280689 A352961

Adjacent sequences: A300380 A300381 A300382 * A300384 A300385 A300386

Keyword

nonn

Author

Gus Wiseman, Mar 04 2018

Status

approved