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A325406
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Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.
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10
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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
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OFFSET
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0,9
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COMMENTS
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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}.
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LINKS
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Table of n, a(n) for n=0..84.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union@@Table[Differences[#, i], {i, 0, Length[#]}]]==k&]], {n, 0, 16}, {k, 0, n}]
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CROSSREFS
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Row sums are A000041.
Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466.
Sequence in context: A144874 A333365 A303065 * A360675 A257900 A039971
Adjacent sequences: A325403 A325404 A325405 * A325407 A325408 A325409
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KEYWORD
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nonn,tabl
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AUTHOR
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Gus Wiseman, May 03 2019
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STATUS
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approved
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