A325267 Number of integer partitions of n with omicron 2.

0, 0, 1, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A304634.

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. We define the omicron of an integer partition to be 0 if the partition is empty, 1 if it is a singleton, and otherwise the second-to-last part of its omega-sequence. 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), and its omicron is 2.

Examples

			The a(1) = 1 through a(8) = 17 partitions:
  (11)  (21)  (22)   (32)    (33)     (43)      (44)
              (31)   (41)    (42)     (52)      (53)
              (211)  (221)   (51)     (61)      (62)
                     (311)   (411)    (322)     (71)
                     (2111)  (2211)   (331)     (332)
                             (3111)   (511)     (422)
                             (21111)  (2221)    (611)
                                      (3211)    (3221)
                                      (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)
		

Crossrefs

Cf. A056239, A112798, A181819, A304634, A304636, A323014, A323023, A325250, A325273, A325277.

Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Switch[#,{},0,{},1,,NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]//First]==2&]],{n,0,30}]