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.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2019
STATUS
approved