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

0, 0, 0, 1, 1, 2, 6, 7, 12, 18, 29, 38, 58, 77, 110, 145, 198, 257, 345, 441, 576, 733, 942, 1184, 1503, 1875, 2352, 2914, 3620, 4454, 5493, 6716, 8221, 10001, 12167, 14723, 17816, 21459, 25836, 30988, 37139, 44365, 52956, 63022, 74934, 88873, 105296, 124469

Offset: 0

Author

Gus Wiseman, Apr 23 2019

Keywords

• nonn

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).

Examples

			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)
		

Crossrefs

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325261, A325277, A325285.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325249 (sum).

Programs

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}]