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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 1 01:12 EDT 2020. Contains 333152 sequences. (Running on oeis4.)