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A325415 Number of distinct sums of omega-sequences of integer partitions of n.

%I A325415 #5 Apr 25 2019 13:30:57
%S A325415 1,1,2,3,4,5,8,8,10,11,13,12,15,14,16,18,18,18,21,20,23,23,24,24,27,
%T A325415 27,28,29,30,30,34,32,34,35,36,37,39,38,40,41,43,42,45,44,46,48,48,48,
%U A325415 51,50,53,53,54,54,57,57,58,59,60,60,64
%N A325415 Number of distinct sums of omega-sequences of integer partitions of n.
%C A325415 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.
%e A325415 The partitions of 9 organized by sum of omega sequence (first column) are:
%e A325415    1: (9)
%e A325415    4: (333)
%e A325415    5: (81) (72) (63) (54)
%e A325415    7: (621) (531) (432)
%e A325415    8: (711) (522) (441)
%e A325415    9: (6111) (3222) (222111)
%e A325415   10: (51111) (33111) (22221) (111111111)
%e A325415   11: (411111)
%e A325415   12: (5211) (4311) (4221) (3321) (3111111) (2211111)
%e A325415   13: (42111) (32211) (21111111)
%e A325415   14: (321111)
%e A325415 There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.
%t A325415 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325415 Table[Length[Union[Total/@omseq/@IntegerPartitions[n]]],{n,0,30}]
%Y A325415 Number of nonzero terms in row n of A325414.
%Y A325415 Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, A325416.
%Y A325415 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 A325415 nonn
%O A325415 0,3
%A A325415 _Gus Wiseman_, Apr 24 2019