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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
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
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 * A334906 A078471 A127406
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved