cp's OEIS Frontend

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

%I A325246 #5 Apr 16 2019 15:27:04
%S A325246 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 A325246 178,202,228,255,286,317,355,390,430,472,519,566,617,670,728,787,852,
%U A325246 916,988,1060,1137,1218,1303,1389,1482,1577,1679,1781,1890,2001,2120
%N A325246 Number of integer partitions of n with adjusted frequency depth equal to their length.
%C A325246 The Heinz numbers of these partitions are given by A325266.
%C A325246 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 A325246 The a(1) = 1 through a(10) = 14 partitions (A = 10):
%e A325246   (1)  (2)   (3)  (4)   (5)     (6)     (7)     (8)      (9)      (A)
%e A325246        (11)       (22)  (2111)  (33)    (421)   (44)     (432)    (55)
%e A325246                                 (321)   (2221)  (431)    (531)    (532)
%e A325246                                 (3111)  (4111)  (521)    (621)    (541)
%e A325246                                                 (5111)   (3222)   (631)
%e A325246                                                 (32111)  (6111)   (721)
%e A325246                                                          (32211)  (3331)
%e A325246                                                          (42111)  (4222)
%e A325246                                                                   (7111)
%e A325246                                                                   (32221)
%e A325246                                                                   (33211)
%e A325246                                                                   (42211)
%e A325246                                                                   (43111)
%e A325246                                                                   (52111)
%t A325246 fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
%t A325246 Table[Length[Select[IntegerPartitions[n],fdadj[#]==Length[#]&]],{n,0,30}]
%Y A325246 Cf. A001222, A008284, A127002, A181819, A182850, A225485, A323014, A323023, A325254, A325266, A325277, A325280, A325281.
%K A325246 nonn
%O A325246 0,3
%A A325246 _Gus Wiseman_, Apr 15 2019