A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353

Offset: 0

Author

Gus Wiseman, May 23 2023

Keywords

• nonn

Comments

Equivalently, n = (length)*(minimum).

Examples

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)
		

Crossrefs

Removing the factor 2 gives A099777.

Taking maximum instead of mean and including odd indices gives A118096.

For length instead of mean and including odd indices we have A237757.

For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.

For median instead of mean we have A361861.

These partitions have ranks A363133.

For maximum instead of minimum we have A363218.

For median instead of minimum we have A363224.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A051293 counts subsets with integer mean.

A067538 counts partitions with integer mean.

A268192 counts partitions by complement size, ranks A326844.

Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

Python

from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

Extensions

a(31)-a(46) from Chai Wah Wu, Sep 21 2023