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A325262 Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers. 4
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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..47.

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

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).

Sequence in context: A190121 A105329 A124319 * A078471 A127406 A092310

Adjacent sequences:  A325259 A325260 A325261 * A325263 A325264 A325265

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

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Last modified August 24 22:41 EDT 2019. Contains 326314 sequences. (Running on oeis4.)