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%I #6 Apr 25 2019 09:32:59
%S 0,0,0,1,1,2,6,7,12,18,29,38,58,77,110,145,198,257,345,441,576,733,
%T 942,1184,1503,1875,2352,2914,3620,4454,5493,6716,8221,10001,12167,
%U 14723,17816,21459,25836,30988,37139,44365,52956,63022,74934,88873,105296,124469
%N Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.
%C 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).
%e The a(3) = 1 through a(9) = 18 partitions:
%e (111) (1111) (2111) (222) (421) (431) (333)
%e (11111) (321) (2221) (521) (432)
%e (2211) (4111) (2222) (531)
%e (3111) (22111) (3311) (621)
%e (21111) (31111) (5111) (3222)
%e (111111) (211111) (22211) (6111)
%e (1111111) (32111) (22221)
%e (41111) (32211)
%e (221111) (33111)
%e (311111) (42111)
%e (2111111) (51111)
%e (11111111) (222111)
%e (321111)
%e (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e (111111111)
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t Table[Length[Select[IntegerPartitions[n],!normQ[omseq[#]]&]],{n,0,30}]
%Y Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.
%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325249 (sum).
%K nonn
%O 0,6
%A _Gus Wiseman_, Apr 23 2019
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