A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 4, 2, 3, 1, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 1, 3, 5, 6, 3, 3, 1, 0, 0, 1, 3, 4, 8, 7, 1, 4, 2, 0, 0, 1, 3, 6, 11, 7, 5, 2, 4, 2, 1, 0, 1, 1, 6, 13, 8, 9, 9, 0, 4, 3, 1, 0, 1, 6, 7, 11, 12, 9

Offset: 0

Author

Gus Wiseman, May 03 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. The distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:

(1,1,2,4)

(0,1,2)

(1,1)

(0)

so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  0
  0  1  2  2  0
  0  1  1  3  2  0
  0  1  4  2  3  1  0
  0  1  1  5  5  2  1  0
  0  1  3  5  6  3  3  1  0
  0  1  3  4  8  7  1  4  2  0
  0  1  3  6 11  7  5  2  4  2  1
  0  1  1  6 13  8  9  9  0  4  3  1
  0  1  6  7 11 12  9 10  8  4  3  2  2
  0  1  1  7 18  9 14 19  5 10  3  5  4  1
  0  1  3  9 17  9 22 20 15  9  7  6  5  4  1
  0  1  4  8 22 11 16 24 22 19 10 11  2  8  7  2
  0  1  4 10 23 15 24 23 27 27 12 14 11  8  8  5  5
Row n = 8 counts the following partitions:
  (8)  (44)        (17)       (116)     (134)   (1133)   (111122)
       (2222)      (26)       (125)     (233)   (11123)
       (11111111)  (35)       (1115)    (1223)  (11222)
                   (224)      (1124)
                   (1111112)  (11114)
                              (111113)
		

Links

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

Crossrefs

Row sums are A000041.

Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466.

Programs

Mathematica

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