A325415 - OEIS

%I #5 Apr 25 2019 13:30:57

%S 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 27,28,29,30,30,34,32,34,35,36,37,39,38,40,41,43,42,45,44,46,48,48,48,

%U 51,50,53,53,54,54,57,57,58,59,60,60,64

%N Number of distinct sums of omega-sequences of integer partitions 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.

%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 There are a total of 11 distinct sums {1,4,5,7,8,9,10,11,12,13,14}, so a(9) = 11.

%t omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];

%t Table[Length[Union[Total/@omseq/@IntegerPartitions[n]]],{n,0,30}]

%Y Number of nonzero terms in row n of A325414.

%Y Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325249, A325277, A325412, A325413, 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