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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. 9

%I

%S 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,

%T 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,

%U 1,6,13,8,9,9,0,4,3,1,0,1,6,7,11,12,9

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

%C 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).

%C 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:

%C (1,1,2,4)

%C (0,1,2)

%C (1,1)

%C (0)

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

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 1 2 0

%e 0 1 2 2 0

%e 0 1 1 3 2 0

%e 0 1 4 2 3 1 0

%e 0 1 1 5 5 2 1 0

%e 0 1 3 5 6 3 3 1 0

%e 0 1 3 4 8 7 1 4 2 0

%e 0 1 3 6 11 7 5 2 4 2 1

%e 0 1 1 6 13 8 9 9 0 4 3 1

%e 0 1 6 7 11 12 9 10 8 4 3 2 2

%e 0 1 1 7 18 9 14 19 5 10 3 5 4 1

%e 0 1 3 9 17 9 22 20 15 9 7 6 5 4 1

%e 0 1 4 8 22 11 16 24 22 19 10 11 2 8 7 2

%e 0 1 4 10 23 15 24 23 27 27 12 14 11 8 8 5 5

%e Row n = 8 counts the following partitions:

%e (8) (44) (17) (116) (134) (1133) (111122)

%e (2222) (26) (125) (233) (11123)

%e (11111111) (35) (1115) (1223) (11222)

%e (224) (1124)

%e (1111112) (11114)

%e (111113)

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

%Y Row sums are A000041.

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

%K nonn,tabl

%O 0,9

%A _Gus Wiseman_, May 03 2019

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Last modified May 31 15:27 EDT 2020. Contains 334748 sequences. (Running on oeis4.)