A360688 -id:A360688 - cp's OEIS frontend

A361392 Number of integer partitions of n whose first differences have mean -1.

0, 0, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 12, 12, 19, 19, 29, 31, 43, 48, 65, 73, 97, 110, 142, 164, 208, 240, 301, 350, 432, 504, 617, 719, 874, 1019, 1228, 1434, 1717, 2001, 2385, 2778, 3292, 3831, 4522, 5252, 6177, 7164, 8392, 9722, 11352, 13125, 15283, 17643

Offset: 0

Gus Wiseman, Mar 13 2023

nonn

These are partitions where the first part minus the last part is the number of parts minus 1.

			The a(3) = 1 through a(11) = 8 partitions:
  (21)  .  (32)   (321)  (43)    (422)   (54)     (442)    (65)
           (311)         (331)   (4211)  (432)    (4321)   (533)
                         (4111)          (4221)   (4411)   (4331)
                                         (4311)   (52111)  (4421)
                                         (51111)           (5222)
                                                           (52211)
                                                           (53111)
                                                           (611111)
For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).

For mean 0 we have A032741.
The 0-appended version is A047993.
For any negative mean we have A144300.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 counts partitions with integer mean, ranks A316413.
A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.
A360614/A360615 gives mean of 0-appended first differences of prime indices.
Cf. A102627, A237363, A237832, A348551, A360688.

Table[Length[Select[IntegerPartitions[n],Mean[Differences[#]]==-1&]],{n,0,30}]

cp's OEIS Frontend

A361392 Number of integer partitions of n whose first differences have mean -1.

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