OFFSET
0,5
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).
EXAMPLE
The a(2) = 1 through a(10) = 11 partitions:
(2) (111) (22) (11111) (33) (43) (53) (54) (64)
(1111) (222) (52) (62) (63) (73)
(111111) (61) (71) (72) (82)
(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[#]==Round[Sqrt[n]]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved