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A325285 Number of integer partitions of n whose omega-sequence has repeated parts. 2
0, 0, 0, 1, 2, 5, 6, 13, 17, 26, 36, 54, 66, 98, 125, 164, 214, 285, 354, 468, 585, 745, 945, 1195, 1477, 1864, 2317, 2867, 3544, 4383, 5348, 6589, 8028, 9778, 11885, 14403, 17362, 20992, 25212, 30239, 36158, 43242, 51408, 61240, 72568, 85989, 101607, 120027 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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), which has repeated parts, so (32211) is counted under a(9).
The Heinz numbers of these partitions are given by A325411.
LINKS
EXAMPLE
The a(3) = 1 through a(8) = 17 partitions:
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2111) (3111) (331) (422)
(21111) (421) (431)
(511) (521)
(2221) (611)
(3211) (3221)
(4111) (4211)
(22111) (5111)
(31111) (22211)
(211111) (32111)
(41111)
(221111)
(311111)
(2111111)
MATHEMATICA
omseq[ptn_List]:=If[ptn=={}, {}, Length/@NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@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), A325414 (omega-sequence sum).
Sequence in context: A028259 A327472 A283684 * A323348 A181314 A027010
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 24 2019
STATUS
approved

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Last modified April 15 10:24 EDT 2024. Contains 371681 sequences. (Running on oeis4.)