OFFSET
0,6
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(3) = 1 through a(9) = 18 partitions:
(111) (1111) (2111) (222) (421) (431) (333)
(11111) (321) (2221) (521) (432)
(2211) (4111) (2222) (531)
(3111) (22111) (3311) (621)
(21111) (31111) (5111) (3222)
(111111) (211111) (22211) (6111)
(1111111) (32111) (22221)
(41111) (32211)
(221111) (33111)
(311111) (42111)
(2111111) (51111)
(11111111) (222111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
(111111111)
MATHEMATICA
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n], !normQ[omseq[#]]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved