A237824 - OEIS

%I #24 Mar 16 2024 15:23:16

%S 1,2,3,4,5,7,7,10,11,13,14,19,18,23,25,29,30,38,37,46,48,54,57,70,69,

%T 80,85,97,100,118,118,137,144,159,168,193,195,220,233,259,268,303,311,

%U 348,367,399,419,469,483,532,560,610,639,704,732,801,841,908,954

%N Number of partitions of n such that 2*(least part) >= greatest part.

%C By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - _Gus Wiseman_, May 14 2023

%H John Tyler Rascoe, <a href="/A237824/b237824.txt">Table of n, a(n) for n = 1..300</a>

%F G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - _John Tyler Rascoe_, Mar 07 2024

%e a(6) = 7 counts these partitions:  6, 42, 33, 222, 2211, 21111, 111111.

%e From _Gus Wiseman_, May 14 2023: (Start)

%e The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (21)   (22)    (32)     (33)      (43)       (44)

%e              (111)  (211)   (221)    (42)      (322)      (53)

%e                     (1111)  (2111)   (222)     (2221)     (332)

%e                             (11111)  (2211)    (22111)    (422)

%e                                      (21111)   (211111)   (2222)

%e                                      (111111)  (1111111)  (22211)

%e                                                           (221111)

%e                                                           (2111111)

%e                                                           (11111111)

%e The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (21)   (22)    (32)     (33)      (43)       (44)

%e              (111)  (31)    (41)     (42)      (52)       (53)

%e                     (1111)  (221)    (51)      (61)       (62)

%e                             (11111)  (222)     (331)      (71)

%e                                      (2211)    (2221)     (332)

%e                                      (111111)  (1111111)  (2222)

%e                                                           (3311)

%e                                                           (22211)

%e                                                           (11111111)

%e (End)

%t z = 60; q[n_] := q[n] = IntegerPartitions[n];

%t Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)

%t Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)

%t Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)

%t Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)

%t Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)

%o (PARI)

%o N=60; x='x+O('x^N);

%o gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));

%o Vec(gf) \\ _John Tyler Rascoe_, Mar 07 2024

%Y Cf. A237821, A118096, A053263.

%Y The complement is counted by A237820, ranks A362982.

%Y For modes instead of middles we have A362619, counted by A171979.

%Y These partitions have ranks A362981.

%Y A000041 counts integer partitions, strict A000009.

%Y A325347 counts partitions with integer median, complement A307683.

%Y Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

%K nonn

%O 1,2

%A _Clark Kimberling_, Feb 16 2014