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%I #8 Apr 25 2019 13:30:44
%S 0,1,3,5,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
%U 53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N Largest sum of the omega-sequence of an integer partition of n.
%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) with sum 13.
%C Appears to contain all nonnegative integers except 2, 4, 6, 7, and 11.
%e The partitions of 9 organized by sum of omega-sequence (first column) are:
%e 1: (9)
%e 4: (333)
%e 5: (81) (72) (63) (54)
%e 7: (621) (531) (432)
%e 8: (711) (522) (441)
%e 9: (6111) (3222) (222111)
%e 10: (51111) (33111) (22221) (111111111)
%e 11: (411111)
%e 12: (5211) (4311) (4221) (3321) (3111111) (2211111)
%e 13: (42111) (32211) (21111111)
%e 14: (321111)
%e The largest term in the first column is 14, so a(9) = 14.
%t omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t Table[Max[Total/@omseq/@IntegerPartitions[n]],{n,0,30}]
%Y Row lengths of A325414.
%Y Cf. A181819, A225486, A323014, A323023, A325238, A325249, A325412, A325415, A325416.
%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 0,3
%A _Gus Wiseman_, Apr 24 2019
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