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