OFFSET
0,3
COMMENTS
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325397.
LINKS
EXAMPLE
The a(1) = 1 through a(8) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(321) (2221) (332)
(111111) (1111111) (431)
(2222)
(11111111)
The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), whose first and second differences are (0,-1,-1) and (-1,0) respectively.
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And@@Table[GreaterEqual@@Differences[#, k], {k, 0, Length[#]}]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved