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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 Table of n, a(n) for n=0..47. 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 Cf. A047966, A181819, A323014, A323023, A325247, A325250, A325260, A325262, A325411. 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 Adjacent sequences: A325282 A325283 A325284 * A325286 A325287 A325288 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.)