A325258 a(1) = 1; otherwise, first differences of Levine's sequence A011784.

1, 1, 1, 1, 3, 7, 28, 171, 2624, 172613, 139584150, 6837485347187, 266437138079023501057, 508009471379222384299345337895696, 37745517525533091954228691786161750063795478326636142, 5347426383812697233786139576220412396732847744407175515852823296919414647252347610750

Offset: 0

Author

Gus Wiseman, Apr 16 2019

Keywords

• nonn

Comments

a(n) is the number of nonnegative integers k such that the maximum adjusted frequency depth among integer partitions of k is n. For example, the a(5) = 7 numbers are 7, 8, 9, 10, 11, 12, and 13.

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is A323014(n). The maximum adjusted frequency depth for partitions of n is A325282(n).

Crossrefs

Run lengths of A325282.

Cf. A011784, A181819, A181821, A182850, A182857, A225485, A225486, A323014, A325238, A325254, A325278, A325280, A325283.

Programs

Mathematica

grw[q_]:=Join@@Table[ConstantArray[i,q[[Length[q]-i+1]]],{i,Length[q]}];
ReplacePart[Differences[Last/@NestList[grw,{1,1},9]],2->1]