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

Author

Gus Wiseman, Sep 13 2019

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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

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