cp's OEIS Frontend

A360688 Number of integer partitions of n with integer median of 0-appended first differences.

%I A360688 #6 Feb 22 2023 08:07:58
%S A360688 1,1,3,4,5,7,12,18,25,32,46,62,79,109,142,189,240,322,405,522,671,853,
%T A360688 1053,1345,1653,2081,2551,3174,3878,4826,5851,7219,8747,10712,12936,
%U A360688 15719,18876,22872,27365,32926,39253,47070,55857,66676,79029,93864,110832
%N A360688 Number of integer partitions of n with integer median of 0-appended first differences.
%C A360688 Includes all partitions of odd length (A027193).
%C A360688 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A360688 The a(1) = 1 through a(8) = 18 partitions:
%e A360688   (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
%e A360688             (21)   (22)    (41)     (42)      (43)       (44)
%e A360688             (111)  (211)   (221)    (222)     (61)       (62)
%e A360688                    (1111)  (311)    (321)     (322)      (332)
%e A360688                            (11111)  (411)     (331)      (422)
%e A360688                                     (21111)   (421)      (431)
%e A360688                                     (111111)  (511)      (521)
%e A360688                                               (3211)     (611)
%e A360688                                               (22111)    (2222)
%e A360688                                               (31111)    (3221)
%e A360688                                               (211111)   (4211)
%e A360688                                               (1111111)  (22211)
%e A360688                                                          (32111)
%e A360688                                                          (41111)
%e A360688                                                          (221111)
%e A360688                                                          (311111)
%e A360688                                                          (2111111)
%e A360688                                                          (11111111)
%e A360688 For example, the partition y = (3,2,2,1) has 0-appended parts (3,2,2,1,0), with differences (1,0,1,1), and the multiset {0,1,1,1} has median 1, so y is counted under a(8).
%t A360688 Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]
%Y A360688 The case of median 0 is A360254, ranks A360558.
%Y A360688 These partitions have ranks A360556, complement A360557.
%Y A360688 A000041 counts integer partitions, strict A000009.
%Y A360688 A008284 counts partitions by number of parts.
%Y A360688 A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A360688 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A360688 Cf. A000975, A027193, A237363, A240219, A360455, A360555.
%K A360688 nonn
%O A360688 1,3
%A A360688 _Gus Wiseman_, Feb 20 2023