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).
EXAMPLE
The a(1) = 1 through a(9) = 15 omega-sequences:
(1) (1) (1) (1) (1) (1) (1) (1) (1)
(21) (31) (21) (51) (21) (71) (21) (31)
(221) (41) (221) (31) (221) (41) (91)
(221) (3221) (61) (331) (81) (221)
(3221) (4221) (221) (3221) (221) (331)
(331) (4221) (331) (621)
(421) (5221) (421) (3221)
(3221) (6221) (3221) (4221)
(4221) (43221) (4221) (5221)
(5221) (5221) (6221)
(6221) (7221)
(7221) (8221)
(43221) (43221)
(53221) (53221)
(63221)
MATHEMATICA
omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];
Table[Length[Union[omseq/@IntegerPartitions[n]]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2019
STATUS
approved