login
A325245
Number of integer partitions of n with adjusted frequency depth 3.
9
0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964
OFFSET
0,6
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). 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(3) = 1 through a(10) = 11 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(41) (51) (52) (62) (63) (73)
(321) (61) (71) (72) (82)
(2211) (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[#]==3&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2019
STATUS
approved