A325411 -id:A325411 - cp's OEIS frontend

A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

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

Offset: 0

Gus Wiseman, Apr 24 2019

nonn

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.

			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)

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).

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],!UnsameQ@@omseq[#]&]],{n,0,30}]

cp's OEIS Frontend

A325285 Number of integer partitions of n whose omega-sequence has repeated parts.

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