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

1, 1, 2, 3, 5, 5, 10, 9, 14, 15, 20, 21, 33, 30, 39, 45, 54, 54, 69, 68, 85, 90, 100, 104, 128, 127, 141, 153, 172, 175, 205, 203, 229, 240, 257, 274, 308, 309, 335, 356, 390, 395, 437, 444, 481, 506, 530, 549, 602, 609, 648, 672, 710, 727, 777, 798, 848, 871

Offset: 0

Author

Gus Wiseman, Apr 24 2019

Keywords

• nonn

Comments

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).

Examples

			The a(1) = 1 through a(9) = 15 omega-sequences:
  (1)  (1)   (1)    (1)     (1)     (1)     (1)      (1)      (1)
       (21)  (31)   (21)    (51)    (21)    (71)     (21)     (31)
             (221)  (41)    (221)   (31)    (221)    (41)     (91)
                    (221)   (3221)  (61)    (331)    (81)     (221)
                    (3221)  (4221)  (221)   (3221)   (221)    (331)
                                    (331)   (4221)   (331)    (621)
                                    (421)   (5221)   (421)    (3221)
                                    (3221)  (6221)   (3221)   (4221)
                                    (4221)  (43221)  (4221)   (5221)
                                    (5221)           (5221)   (6221)
                                                     (6221)   (7221)
                                                     (7221)   (8221)
                                                     (43221)  (43221)
                                                     (53221)  (53221)
                                                              (63221)
		

Crossrefs

Cf. A181819, A225486, A323014, A323023, A325238, A325248, A325277, A325413, 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).

Programs

Mathematica

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Union[omseq/@IntegerPartitions[n]]],{n,0,30}]