The On-Line Encyclopedia of Integer Sequences (OEIS)

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A325262

Number of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.
(history; published version)

#6 by Susanna Cuyler at Thu Apr 25 09:32:59 EDT 2019

STATUS

proposed

approved

#5 by Gus Wiseman at Thu Apr 25 09:13:09 EDT 2019

STATUS

editing

proposed

#4 by Gus Wiseman at Thu Apr 25 09:12:36 EDT 2019

CROSSREFS

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

#3 by Gus Wiseman at Tue Apr 23 00:59:36 EDT 2019

CROSSREFS

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

#2 by Gus Wiseman at Tue Apr 23 00:32:56 EDT 2019

NAME

allocated for Gus WisemanNumber of integer partitions of n whose omega-sequence does not cover an initial interval of positive integers.

DATA

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

0,6

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

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 (length/frequency depth).

KEYWORD

allocated

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

editing

#1 by Gus Wiseman at Sun Apr 14 10:31:36 EDT 2019

NAME

allocated for Gus Wiseman

KEYWORD

allocated

STATUS

approved