

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
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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 kth differences for k > 0 are the differences of the (k1)th differences. The distinct differences of any degree are the union of the kth differences for all k >= 0. For example, the kth 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}.


LINKS

Table of n, a(n) for n=0..84.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.


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

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


KEYWORD

nonn,tabl


AUTHOR

Gus Wiseman, May 03 2019


STATUS

approved



