cp's OEIS Frontend

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

%I A361862 #9 Apr 12 2023 11:08:26
%S A361862 0,0,0,1,0,1,0,3,2,2,0,7,0,3,6,10,0,13,0,17,10,5,0,40,12,6,18,34,0,62,
%T A361862 0,50,24,8,60,125,0,9,32,169,0,165,0,95,176,11,0,373,114,198,54,143,0,
%U A361862 384,254,574,66,14,0,1090,0,15,748,633,448,782,0,286
%N A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).
%C A361862 In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.
%e A361862 The a(4) = 1 through a(12) = 7 partitions:
%e A361862   (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
%e A361862                      (3221)  (522)  (33211)     (642)
%e A361862                      (3311)                     (4431)
%e A361862                                                 (5322)
%e A361862                                                 (322221)
%e A361862                                                 (332211)
%e A361862                                                 (333111)
%e A361862 The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
%e A361862   o o o o
%e A361862   o o o o
%e A361862   o o o .
%e A361862   o . . .
%e A361862 Both the rectangle from the left and the complement have size 4.
%t A361862 Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]
%Y A361862 Positions of zeros are 1 and A000040.
%Y A361862 For length instead of mean we have A237832.
%Y A361862 For minimum instead of mean we have A118096.
%Y A361862 These partitions have ranks A362047.
%Y A361862 A000041 counts integer partitions, strict A000009.
%Y A361862 A008284 counts partitions by length, A058398 by mean.
%Y A361862 A067538 counts partitions with integer mean.
%Y A361862 A097364 counts partitions by (maximum) - (minimum).
%Y A361862 A243055 subtracts the least prime index from the greatest.
%Y A361862 A326844 gives the diagram complement size of Heinz partition.
%Y A361862 Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.
%K A361862 nonn
%O A361862 1,8
%A A361862 _Gus Wiseman_, Apr 10 2023