A325413 Largest sum of the omega-sequence of an integer partition of n.

0, 1, 3, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 0

Author

Gus Wiseman, Apr 24 2019

Keywords

• nonn

Comments

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13.

Appears to contain all nonnegative integers except 2, 4, 6, 7, and 11.

Examples

			The partitions of 9 organized by sum of omega-sequence (first column) are:
   1: (9)
   4: (333)
   5: (81) (72) (63) (54)
   7: (621) (531) (432)
   8: (711) (522) (441)
   9: (6111) (3222) (222111)
  10: (51111) (33111) (22221) (111111111)
  11: (411111)
  12: (5211) (4311) (4221) (3321) (3111111) (2211111)
  13: (42111) (32211) (21111111)
  14: (321111)
The largest term in the first column is 14, so a(9) = 14.
		

Crossrefs

Row lengths of A325414.

Cf. A181819, A225486, A323014, A323023, A325238, A325249, A325412, A325415, A325416.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

Programs

Mathematica

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Max[Total/@omseq/@IntegerPartitions[n]],{n,0,30}]