A327472 Number of integer partitions of n not containing their mean.

1, 0, 0, 1, 2, 5, 6, 13, 16, 25, 34, 54, 56, 99, 121, 154, 201, 295, 324, 488, 541, 725, 957, 1253, 1292, 1892, 2356, 2813, 3378, 4563, 4838, 6840, 7686, 9600, 12076, 14180, 15445, 21635, 25627, 29790, 33309, 44581, 48486, 63259, 70699, 82102, 104553, 124752

Offset

0,5

Links

Table of n, a(n) for n=0..47.

Example

The a(3) = 1 through a(8) = 16 partitions not containing their mean:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (411)    (61)      (71)
               (311)   (2211)   (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3311)
                                (4111)    (5111)
                                (22111)   (22211)
                                (31111)   (32111)
                                (211111)  (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !MemberQ[#, Mean[#]]&]], {n, 0, 20}]

Prog

(Python)
from sympy.utilities.iterables import partitions
def A327472(n): return sum(1 for s, p in partitions(n, size=True) if n%s or n//s not in p) if n else 1 # Chai Wah Wu, Sep 21 2023

Crossrefs

The Heinz numbers of these partitions are A327476.

Partitions with their mean are A237984.

Subsets without their mean are A327471.

Subsets with n but without their mean are A327477.

Strict partitions without their mean are A240851.

Cf. A007865, A067538, A067538, A102627, A114639, A324756, A327473.

Sequence in context: A098376 A342569 A028259 * A283684 A325285 A323348

Adjacent sequences: A327469 A327470 A327471 * A327473 A327474 A327475

Keyword

nonn

Author

Gus Wiseman, Sep 13 2019

Status

approved