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Number of integer partitions of n with adjusted frequency depth equal to their length.
7

%I #5 Apr 16 2019 15:27:04

%S 1,1,2,1,2,2,4,4,6,8,14,15,21,26,34,42,51,60,74,86,102,117,137,155,

%T 178,202,228,255,286,317,355,390,430,472,519,566,617,670,728,787,852,

%U 916,988,1060,1137,1218,1303,1389,1482,1577,1679,1781,1890,2001,2120

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

%C The Heinz numbers of these partitions are given by A325266.

%C 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.

%e The a(1) = 1 through a(10) = 14 partitions (A = 10):

%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A)

%e (11) (22) (2111) (33) (421) (44) (432) (55)

%e (321) (2221) (431) (531) (532)

%e (3111) (4111) (521) (621) (541)

%e (5111) (3222) (631)

%e (32111) (6111) (721)

%e (32211) (3331)

%e (42111) (4222)

%e (7111)

%e (32221)

%e (33211)

%e (42211)

%e (43111)

%e (52111)

%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];

%t Table[Length[Select[IntegerPartitions[n],fdadj[#]==Length[#]&]],{n,0,30}]

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 15 2019