A237821 - OEIS

%I #14 May 27 2023 10:33:02

%S 0,0,1,2,4,7,11,16,25,35,48,68,92,123,164,216,282,367,471,604,769,975,

%T 1225,1542,1924,2395,2968,3669,4514,5547,6781,8280,10071,12229,14796,

%U 17881,21537,25902,31066,37206,44443,53021,63098,74995,88946,105350,124533

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

%C By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - _Gus Wiseman_, May 15 2023

%F G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - _Seiichi Manyama_, May 27 2023

%e a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.

%e From _Gus Wiseman_, May 15 2023: (Start)

%e The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:

%e   (21)  (31)   (41)    (42)     (52)

%e         (211)  (221)   (51)     (61)

%e                (311)   (321)    (331)

%e                (2111)  (411)    (421)

%e                        (2211)   (511)

%e                        (3111)   (2221)

%e                        (21111)  (3211)

%e                                 (4111)

%e                                 (22111)

%e                                 (31111)

%e                                 (211111)

%e The a(3) = 1 through a(8) = 16 partitions with different median from maximum:

%e   (21)  (31)   (32)    (42)     (43)

%e         (211)  (41)    (51)     (52)

%e                (311)   (321)    (61)

%e                (2111)  (411)    (322)

%e                        (2211)   (421)

%e                        (3111)   (511)

%e                        (21111)  (3211)

%e                                 (4111)

%e                                 (22111)

%e                                 (31111)

%e                                 (211111)

%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}] (* A237824 *)

%Y The complement is counted by A053263, ranks A081306.

%Y These partitions have ranks A069900.

%Y The case of equality is A118096.

%Y For < instead of <= we have A237820, ranks A362982.

%Y For >= instead of <= we have A237824, ranks A362981.

%Y The conjugate partitions have ranks A362980.

%Y A000041 counts integer partitions, strict A000009.

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

%Y Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Feb 16 2014