A238479 Number of partitions of n whose median is not a part.

0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 13, 18, 23, 30, 40, 50, 64, 83, 104, 131, 166, 206, 256, 320, 394, 485, 598, 730, 891, 1088, 1318, 1596, 1932, 2326, 2797, 3360, 4020, 4804, 5735, 6824, 8108, 9624, 11392, 13468, 15904, 18737, 22048, 25914, 30400, 35619, 41686

Offset: 1

Author

Clark Kimberling, Feb 27 2014

Keywords

• nonn
• easy

Comments

Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.

Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023

Examples

			a(6) counts these partitions:  51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.
		

Links

• A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

Crossrefs

The complement is A238478, ranks A362618.

For mean instead of median we have A327472, complement A237984.

These partitions have ranks A362617.

A000041 counts integer partitions, even-length A027187.

A325347 counts partitions with integer median, complement A307683.

A359893/A359901/A359902 count partitions by median.

A359908 ranks partitions with integer median, complement A359912.

Cf. A002865, A008284, A053263, A238480, A238481, A240219, A359907, A360686, A360687, A360952.

Programs

Mathematica

Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]

PARI

my(q='q+O('q^50)); concat([0,0], Vec(sum(n=1,17,q^(3*n)/prod(k=1,2*n,1-q^k)))) \\ David Radcliffe, Jun 25 2025

Python

from sympy.utilities.iterables import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p,default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023

Formula

a(n) = A000041(n) - A238478(n).

For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014

From Jeremy Lovejoy, Sep 29 2022: (Start)

G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).

a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)

a(n) = A087897(2*n) = A035294(n) - A078408(n-1). - Mathew Englander, May 20 2023