cp's OEIS Frontend

A325412 Number of distinct omega-sequences of integer partitions of n.

%I A325412 #5 Apr 25 2019 09:33:32
%S A325412 1,1,2,3,5,5,10,9,14,15,20,21,33,30,39,45,54,54,69,68,85,90,100,104,
%T A325412 128,127,141,153,172,175,205,203,229,240,257,274,308,309,335,356,390,
%U A325412 395,437,444,481,506,530,549,602,609,648,672,710,727,777,798,848,871
%N A325412 Number of distinct omega-sequences of integer partitions of n.
%C A325412 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).
%e A325412 The a(1) = 1 through a(9) = 15 omega-sequences:
%e A325412   (1)  (1)   (1)    (1)     (1)     (1)     (1)      (1)      (1)
%e A325412        (21)  (31)   (21)    (51)    (21)    (71)     (21)     (31)
%e A325412              (221)  (41)    (221)   (31)    (221)    (41)     (91)
%e A325412                     (221)   (3221)  (61)    (331)    (81)     (221)
%e A325412                     (3221)  (4221)  (221)   (3221)   (221)    (331)
%e A325412                                     (331)   (4221)   (331)    (621)
%e A325412                                     (421)   (5221)   (421)    (3221)
%e A325412                                     (3221)  (6221)   (3221)   (4221)
%e A325412                                     (4221)  (43221)  (4221)   (5221)
%e A325412                                     (5221)           (5221)   (6221)
%e A325412                                                      (6221)   (7221)
%e A325412                                                      (7221)   (8221)
%e A325412                                                      (43221)  (43221)
%e A325412                                                      (53221)  (53221)
%e A325412                                                               (63221)
%t A325412 omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
%t A325412 Table[Length[Union[omseq/@IntegerPartitions[n]]],{n,0,30}]
%Y A325412 Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, A325415.
%Y A325412 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 A325412 nonn
%O A325412 0,3
%A A325412 _Gus Wiseman_, Apr 24 2019