cp's OEIS Frontend

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

%I A238479 #52 Jun 26 2025 07:57:49
%S A238479 0,0,1,1,2,3,4,5,8,10,13,18,23,30,40,50,64,83,104,131,166,206,256,320,
%T A238479 394,485,598,730,891,1088,1318,1596,1932,2326,2797,3360,4020,4804,
%U A238479 5735,6824,8108,9624,11392,13468,15904,18737,22048,25914,30400,35619,41686
%N A238479 Number of partitions of n whose median is not a part.
%C A238479 Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.
%C A238479 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
%H A238479 A. Blecher and A. Knopfmacher, <a href="http://doi.org/10.1007/s11139-022-00551-x">Fixed points and matching points in partitions</a>, Ramanujan J. 58 (2022), 23-41.
%F A238479 a(n) = A000041(n) - A238478(n).
%F A238479 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
%F A238479 From _Jeremy Lovejoy_, Sep 29 2022: (Start)
%F A238479 G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
%F A238479 a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
%F A238479 a(n) = A087897(2*n) = A035294(n) - A078408(n-1). - _Mathew Englander_, May 20 2023
%e A238479 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.
%t A238479 Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
%t A238479 (* also *)
%t A238479 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]
%o A238479 (Python)
%o A238479 from sympy.utilities.iterables import partitions
%o A238479 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
%o A238479 (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
%Y A238479 The complement is A238478, ranks A362618.
%Y A238479 For mean instead of median we have A327472, complement A237984.
%Y A238479 These partitions have ranks A362617.
%Y A238479 A000041 counts integer partitions, even-length A027187.
%Y A238479 A325347 counts partitions with integer median, complement A307683.
%Y A238479 A359893/A359901/A359902 count partitions by median.
%Y A238479 A359908 ranks partitions with integer median, complement A359912.
%Y A238479 Cf. A002865, A008284, A053263, A238480, A238481, A240219, A359907, A360686, A360687, A360952.
%K A238479 nonn,easy
%O A238479 1,5
%A A238479 _Clark Kimberling_, Feb 27 2014