cp's OEIS Frontend

A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.

%I A361854 #7 Mar 31 2023 05:01:14
%S A361854 0,0,0,0,0,0,0,0,1,0,0,2,0,1,2,2,0,5,0,6,3,5,0,11,6,8,7,10,0,36,0,14,
%T A361854 16,16,29,43,0,21,36,69,0,97,0,35,138,33,0,150,61,137,134,74,0,231,
%U A361854 134,265,229,56,0,650,0,65,749,267,247,533,0,405,565
%N A361854 Number of strict integer partitions of n such that (length) * (maximum) = 2n.
%C A361854 Also strict partitions satisfying (maximum) = 2*(mean).
%C A361854 These are strict partitions where both the diagram and its complement (see example) have size n.
%e A361854 The a(n) strict partitions for selected n (A..E = 10..14):
%e A361854   n=9:  n=12:  n=14:  n=15:  n=16:  n=18:  n=20:  n=21:  n=22:
%e A361854 --------------------------------------------------------------
%e A361854   621   831    7421   A32    8431   C42    A532   E43    B542
%e A361854         6321          A41    8521   C51    A541   E52    B632
%e A361854                                     9432   A631   E61    B641
%e A361854                                     9531   A721          B731
%e A361854                                     9621   85421         B821
%e A361854                                            86321
%e A361854 The a(20) = 6 strict partitions are: (10,7,2,1), (10,6,3,1), (10,5,4,1), (10,5,3,2), (8,6,3,2,1), (8,5,4,2,1).
%e A361854 The strict partition y = (8,5,4,2,1) has diagram:
%e A361854   o o o o o o o o
%e A361854   o o o o o . . .
%e A361854   o o o o . . . .
%e A361854   o o . . . . . .
%e A361854   o . . . . . . .
%e A361854 Since the partition and its complement (shown in dots) have the same size, y is counted under a(20).
%t A361854 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[#]*Max@@#==2n&]],{n,30}]
%Y A361854 For minimum instead of mean we have A241035, non-strict A118096.
%Y A361854 For length instead of mean we have A241087, non-strict A237753.
%Y A361854 For median instead of mean we have A361850, non-strict A361849.
%Y A361854 The non-strict version is A361853.
%Y A361854 These partitions have ranks A361855 /\ A005117.
%Y A361854 A000041 counts integer partitions, strict A000009.
%Y A361854 A008284 counts partitions by length, A058398 by mean.
%Y A361854 A008289 counts strict partitions by length.
%Y A361854 A102627 counts strict partitions with integer mean, non-strict A067538.
%Y A361854 A116608 counts partitions by number of distinct parts.
%Y A361854 A268192 counts partitions by complement size, ranks A326844.
%Y A361854 Cf. A111907, A237755, A240850, A326849 A359897, A360068, A360071, A360243, A361848, A361851, A361852, A361906.
%K A361854 nonn
%O A361854 1,12
%A A361854 _Gus Wiseman_, Mar 29 2023