A325246 Number of integer partitions of n with adjusted frequency depth equal to their length.

1, 1, 2, 1, 2, 2, 4, 4, 6, 8, 14, 15, 21, 26, 34, 42, 51, 60, 74, 86, 102, 117, 137, 155, 178, 202, 228, 255, 286, 317, 355, 390, 430, 472, 519, 566, 617, 670, 728, 787, 852, 916, 988, 1060, 1137, 1218, 1303, 1389, 1482, 1577, 1679, 1781, 1890, 2001, 2120

Offset

0,3

Comments

The Heinz numbers of these partitions are given by A325266.

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(1) = 1 through a(10) = 14 partitions (A = 10):
  (1)  (2)   (3)  (4)   (5)     (6)     (7)     (8)      (9)      (A)
       (11)       (22)  (2111)  (33)    (421)   (44)     (432)    (55)
                                (321)   (2221)  (431)    (531)    (532)
                                (3111)  (4111)  (521)    (621)    (541)
                                                (5111)   (3222)   (631)
                                                (32111)  (6111)   (721)
                                                         (32211)  (3331)
                                                         (42111)  (4222)
                                                                  (7111)
                                                                  (32221)
                                                                  (33211)
                                                                  (42211)
                                                                  (43111)
                                                                  (52111)

Mathematica

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

Crossrefs

Cf. A001222, A008284, A127002, A181819, A182850, A225485, A323014, A323023, A325254, A325266, A325277, A325280, A325281.

Sequence in context: A325707 A026832 A225044 * A193691 A089408 A350287

Adjacent sequences: A325243 A325244 A325245 * A325247 A325248 A325249

Keyword

nonn

Author

Gus Wiseman, Apr 15 2019

Status

approved