

A325413


Largest sum of the omegasequence 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
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OFFSET

0,3


COMMENTS

The omegasequence 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 omegasequence 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 omegasequence (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 (secondtolast omega), A225485 or A325280 (frequency depth), A325414 (omegasequence 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



