cp's OEIS Frontend

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

%I A361853 #6 Mar 31 2023 05:01:23
%S A361853 0,0,0,0,0,2,0,1,2,4,0,10,0,8,16,10,0,31,0,44,44,20,0,92,50,28,98,154,
%T A361853 0,266,0,154,194,48,434,712,0,60,348,910,0,1198,0,1120,2138,88,0,2428,
%U A361853 1300,1680,912,2506,0,4808,4800,5968,1372,140,0,14820,0,160
%N A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.
%C A361853 Also partitions satisfying (maximum) = 2*(mean).
%C A361853 These are partitions whose diagram has the same size as its complement (see example).
%e A361853 The a(6) = 2 through a(12) = 10 partitions:
%e A361853   (411)   .  (4211)  (621)     (5221)   .  (822)
%e A361853   (3111)             (321111)  (5311)      (831)
%e A361853                                (42211)     (6222)
%e A361853                                (43111)     (6321)
%e A361853                                            (6411)
%e A361853                                            (422211)
%e A361853                                            (432111)
%e A361853                                            (441111)
%e A361853                                            (32211111)
%e A361853                                            (33111111)
%e A361853 The partition y = (6,4,1,1) has diagram:
%e A361853   o o o o o o
%e A361853   o o o o . .
%e A361853   o . . . . .
%e A361853   o . . . . .
%e A361853 Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).
%t A361853 Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#==2n&]],{n,30}]
%Y A361853 For minimum instead of mean we have A118096.
%Y A361853 For length instead of mean we have A237753.
%Y A361853 For median instead of mean we have A361849, ranks A361856.
%Y A361853 This is the equal case of A361851, unequal case A361852.
%Y A361853 The strict case is A361854.
%Y A361853 These partitions have ranks A361855.
%Y A361853 This is the equal case of A361906, unequal case A361907.
%Y A361853 A000041 counts integer partitions, strict A000009.
%Y A361853 A008284 counts partitions by length, A058398 by mean.
%Y A361853 A051293 counts subsets with integer mean.
%Y A361853 A067538 counts partitions with integer mean.
%Y A361853 A268192 counts partitions by complement size, ranks A326844.
%Y A361853 Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.
%K A361853 nonn
%O A361853 1,6
%A A361853 _Gus Wiseman_, Mar 29 2023