cp's OEIS Frontend

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

%I A325285 #8 Aug 22 2019 09:54:38
%S A325285 0,0,0,1,2,5,6,13,17,26,36,54,66,98,125,164,214,285,354,468,585,745,
%T A325285 945,1195,1477,1864,2317,2867,3544,4383,5348,6589,8028,9778,11885,
%U A325285 14403,17362,20992,25212,30239,36158,43242,51408,61240,72568,85989,101607,120027
%N A325285 Number of integer partitions of n whose omega-sequence has repeated parts.
%C A325285 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).
%C A325285 The Heinz numbers of these partitions are given by A325411.
%e A325285 The a(3) = 1 through a(8) = 17 partitions:
%e A325285   (21)  (31)   (32)    (42)     (43)      (53)
%e A325285         (211)  (41)    (51)     (52)      (62)
%e A325285                (221)   (321)    (61)      (71)
%e A325285                (311)   (411)    (322)     (332)
%e A325285                (2111)  (3111)   (331)     (422)
%e A325285                        (21111)  (421)     (431)
%e A325285                                 (511)     (521)
%e A325285                                 (2221)    (611)
%e A325285                                 (3211)    (3221)
%e A325285                                 (4111)    (4211)
%e A325285                                 (22111)   (5111)
%e A325285                                 (31111)   (22211)
%e A325285                                 (211111)  (32111)
%e A325285                                           (41111)
%e A325285                                           (221111)
%e A325285                                           (311111)
%e A325285                                           (2111111)
%t A325285 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325285 Table[Length[Select[IntegerPartitions[n],!UnsameQ@@omseq[#]&]],{n,0,30}]
%Y A325285 Cf. A047966, A181819, A323014, A323023, A325247, A325250, A325260, A325262, A325411.
%Y A325285 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).
%K A325285 nonn
%O A325285 0,5
%A A325285 _Gus Wiseman_, Apr 24 2019