A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

1, 1, 2, 3, 5, 6, 10, 11, 15, 19, 24, 25, 36, 37, 43, 54, 63, 64, 80, 81, 100, 113, 122, 123, 151, 166, 178, 195, 217, 218, 269, 270, 295, 316, 332, 372, 424, 425, 447, 472, 547, 550, 616, 617, 659, 750, 777, 782, 862, 885, 995, 1032, 1083, 1090, 1176, 1275

Offset: 0

Author

Gus Wiseman, May 02 2019

Keywords

• nonn

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 A325400.

Examples

			The a(1) = 1 through a(8) = 15 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (112)   (113)    (33)      (34)       (35)
                    (1111)  (1112)   (114)     (115)      (44)
                            (11111)  (123)     (124)      (116)
                                     (222)     (223)      (125)
                                     (1113)    (1114)     (224)
                                     (11112)   (11113)    (1115)
                                     (111111)  (111112)   (1124)
                                               (1111111)  (2222)
                                                          (11114)
                                                          (111113)
                                                          (1111112)
                                                          (11111111)
		

Links

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

Crossrefs

Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

Programs

Mathematica

Table[Length[Select[Sort/@IntegerPartitions[n],And@@Table[OrderedQ[Differences[#,k]],{k,0,Length[#]}]&]],{n,0,30}]