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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.
10
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
0,9
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}.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union@@Table[Differences[#, i], {i, 0, Length[#]}]]==k&]], {n, 0, 16}, {k, 0, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 03 2019
STATUS
approved