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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
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 (list; table; graph; refs; listen; history; text; internal format)
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}.

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 * A257900 A039971 A205593

Adjacent sequences:  A325403 A325404 A325405 * A325407 A325408 A325409

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, May 03 2019

STATUS

approved

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Last modified March 31 19:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)