A325245 Number of integer partitions of n with adjusted frequency depth 3.

0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964

Offset

0,6

Comments

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014.

Links

Table of n, a(n) for n=0..54.

Example

The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)   (53)    (54)      (64)
              (41)  (51)    (52)   (62)    (63)      (73)
                    (321)   (61)   (71)    (72)      (82)
                    (2211)  (421)  (431)   (81)      (91)
                                   (521)   (432)     (532)
                                   (3311)  (531)     (541)
                                           (621)     (631)
                                           (222111)  (721)
                                                     (3322)
                                                     (4321)
                                                     (4411)

Mathematica

fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n], fdadj[#]==3&]], {n, 0, 30}]

Crossrefs

Column k = 3 of A225485 and A325280.

Cf. A008284, A047966, A116608, A127002, A181819, A182850, A323014, A323023, A325239, A325246, A325254, A325268, A325280.

Sequence in context: A339905 A215205 A298043 * A241064 A292671 A210948

Adjacent sequences: A325242 A325243 A325244 * A325246 A325247 A325248

Keyword

nonn

Author

Gus Wiseman, Apr 15 2019

Status

approved