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A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

%I A325262 #6 Apr 25 2019 09:32:59
%S A325262 0,0,0,1,1,2,6,7,12,18,29,38,58,77,110,145,198,257,345,441,576,733,
%T A325262 942,1184,1503,1875,2352,2914,3620,4454,5493,6716,8221,10001,12167,
%U A325262 14723,17816,21459,25836,30988,37139,44365,52956,63022,74934,88873,105296,124469
%N A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.
%C A325262 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 A325262 The a(3) = 1 through a(9) = 18 partitions:
%e A325262   (111)  (1111)  (2111)   (222)     (421)      (431)       (333)
%e A325262                  (11111)  (321)     (2221)     (521)       (432)
%e A325262                           (2211)    (4111)     (2222)      (531)
%e A325262                           (3111)    (22111)    (3311)      (621)
%e A325262                           (21111)   (31111)    (5111)      (3222)
%e A325262                           (111111)  (211111)   (22211)     (6111)
%e A325262                                     (1111111)  (32111)     (22221)
%e A325262                                                (41111)     (32211)
%e A325262                                                (221111)    (33111)
%e A325262                                                (311111)    (42111)
%e A325262                                                (2111111)   (51111)
%e A325262                                                (11111111)  (222111)
%e A325262                                                            (321111)
%e A325262                                                            (411111)
%e A325262                                                            (2211111)
%e A325262                                                            (3111111)
%e A325262                                                            (21111111)
%e A325262                                                            (111111111)
%t A325262 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325262 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325262 Table[Length[Select[IntegerPartitions[n],!normQ[omseq[#]]&]],{n,0,30}]
%Y A325262 Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.
%Y A325262 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 A325262 nonn
%O A325262 0,6
%A A325262 _Gus Wiseman_, Apr 23 2019