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A350839
Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.
16
0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694
OFFSET
0,7
COMMENTS
We define a difference of a partition to be a difference of two adjacent parts.
EXAMPLE
The a(5) = 1 through a(10) = 17 partitions:
(311) (411) (511) (422) (522) (622)
(3111) (4111) (611) (711) (811)
(31111) (3311) (4221) (4222)
(4211) (4311) (4411)
(5111) (5211) (5221)
(41111) (6111) (5311)
(311111) (33111) (6211)
(42111) (7111)
(51111) (42211)
(411111) (43111)
(3111111) (52111)
(61111)
(331111)
(421111)
(511111)
(4111111)
(31111111)
MATHEMATICA
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], (Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]], {n, 0, 30}]
CROSSREFS
Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Sequence in context: A216825 A094066 A330070 * A161921 A060341 A114345
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 24 2022
STATUS
approved