

A325415


Number of distinct sums of omegasequences of integer partitions of n.


4



1, 1, 2, 3, 4, 5, 8, 8, 10, 11, 13, 12, 15, 14, 16, 18, 18, 18, 21, 20, 23, 23, 24, 24, 27, 27, 28, 29, 30, 30, 34, 32, 34, 35, 36, 37, 39, 38, 40, 41, 43, 42, 45, 44, 46, 48, 48, 48, 51, 50, 53, 53, 54, 54, 57, 57, 58, 59, 60, 60, 64
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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.


LINKS

Table of n, a(n) for n=0..60.


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)
There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.


MATHEMATICA

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


CROSSREFS

Number of nonzero terms in row n of A325414.
Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, 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: A297457 A297694 A297637 * A316496 A100882 A171979
Adjacent sequences: A325412 A325413 A325414 * A325416 A325417 A325418


KEYWORD

nonn


AUTHOR

Gus Wiseman, Apr 24 2019


STATUS

approved



