A325353 Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.

1, 1, 2, 3, 4, 5, 7, 7, 9, 11, 12, 13, 17, 16, 19, 23, 23, 24, 30, 29, 35, 37, 37, 40, 49, 47, 51, 56, 59, 61, 73, 65, 75, 80, 84, 91, 99, 91, 103, 112, 120, 114, 132, 126, 143, 154, 147, 152, 175, 169, 190, 187, 194, 198, 226, 225, 231, 236, 246, 256, 293

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

Table of n, a(n) for n=0..60.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

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

Cf. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.

Sequence in context: A341156 A352921 A341141 * A117174 A320466 A342542

Adjacent sequences: A325350 A325351 A325352 * A325354 A325355 A325356

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved