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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2019
STATUS
approved