A325466 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 5, 4, 0, 1, 0, 0, 1, 4, 6, 3, 0, 1, 0, 0, 1, 6, 6, 4, 3, 1, 1, 0, 0, 1, 6, 10, 4, 2, 4, 1, 2, 0, 0, 1, 7, 12, 8, 3, 3, 4, 1, 2, 1, 0, 1, 6, 13, 11, 2, 11, 3, 4, 0, 3, 1, 1, 1, 10, 16, 7, 10, 10

Offset: 0

Author

Gus Wiseman, May 04 2019

Keywords

• nonn
• tabl

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.

Examples

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  3  1  0  0
  1  3  2  1  0  0
  1  5  4  0  1  0  0
  1  4  6  3  0  1  0  0
  1  6  6  4  3  1  1  0  0
  1  6 10  4  2  4  1  2  0  0
  1  7 12  8  3  3  4  1  2  1  0
  1  6 13 11  2 11  3  4  0  3  1  1
  1 10 16  7 10 10  6  6  5  1  1  2  1
  1  7 18 14  7 16 11  6  4  8  0  5  0  1
  1  9 20 18 10 20 13 10 10  4  5  5  2  2  2
  1 10 26 18 10 24 13 19 13 10  6  6  2  8  1  2
  1 11 25 24 16 28 19 24 14 15  9 10  9  5  2  7  1
Row 7 counts the following reversed partitions (empty columns not shown):
  (7)  (16)       (115)     (133)   (11122)
       (25)       (124)     (1123)
       (34)       (223)     (1222)
       (1111111)  (1114)
                  (11113)
                  (111112)
Row 9 counts the following reversed partitions (empty columns not shown):
(9)  (18)         (117)       (126)    (1125)   (1134)    (11223)  (111222)
     (27)         (135)       (144)    (11124)  (1224)             (1111122)
     (36)         (225)       (1233)            (11133)
     (45)         (234)       (12222)           (111123)
     (333)        (1116)
     (111111111)  (2223)
                  (11115)
                  (111114)
                  (1111113)
                  (11111112)
		

Crossrefs

Row sums are A000041. Column k = 1 is A088922.

Cf. A098859, A279945, A325242, A325324, A325325, A325349, A325404, A325405, A325406, A325468.

Programs

Mathematica

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,1,Length[#]}]]==k&]],{n,0,16},{k,0,n}]