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A325413 Largest sum of the omega-sequence of an integer partition of n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

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.

MATHEMATICA

omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];

Table[Max[Total/@omseq/@IntegerPartitions[n]], {n, 0, 30}]

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).

Sequence in context: A157291 A107820 A191931 * A328066 A064186 A271048

Adjacent sequences:  A325410 A325411 A325412 * A325414 A325415 A325416

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 24 2019

STATUS

approved

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Last modified October 22 13:35 EDT 2019. Contains 328318 sequences. (Running on oeis4.)