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A333213 Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n. 19
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 4, 1, 1, 0, 3, 6, 5, 1, 1, 0, 4, 10, 10, 6, 1, 1, 0, 5, 17, 20, 13, 7, 1, 1, 0, 6, 27, 38, 31, 16, 8, 1, 1, 0, 8, 40, 69, 67, 42, 19, 9, 1, 1, 0, 10, 58, 123, 132, 101, 54, 22, 10, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A composition of n is a finite sequence of positive integers summing to n.

Also the number of compositions of n with k + 1 maximal strictly decreasing subsequences.

Also the number of compositions of n with k adjacent terms that are equal or decreasing (weak descents).

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

Triangle begins:

   1

   0   1

   0   1   1

   0   2   1   1

   0   2   4   1   1

   0   3   6   5   1   1

   0   4  10  10   6   1   1

   0   5  17  20  13   7   1   1

   0   6  27  38  31  16   8   1   1

   0   8  40  69  67  42  19   9   1   1

   0  10  58 123 132 101  54  22  10   1   1

   0  12  86 202 262 218 139  67  25  11   1   1

   0  15 121 332 484 467 324 182  81  28  12   1   1

Row n = 6 counts the following compositions:

  (6)    (15)    (114)   (1113)   (11112)  (111111)

  (42)   (24)    (123)   (1122)

  (51)   (33)    (222)   (11121)

  (321)  (132)   (1131)  (11211)

         (141)   (1212)  (12111)

         (213)   (1221)  (21111)

         (231)   (1311)

         (312)   (2112)

         (411)   (2211)

         (2121)  (3111)

MATHEMATICA

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Split[#, #1>#2&]]==k&]], {n, 0, 12}, {k, 0, n}]

CROSSREFS

Compositions by length are A007318.

The case of reversed partitions (instead of compositions) is A008284.

The version counting equal adjacencies is A106356.

The case of partitions (instead of compositions) is A133121.

The version counting unequal adjacencies is A238279.

The strict/strong version is A238343.

Cf. A072704, A107429, A124764, A124769, A329744, A332875, A333230.

Sequence in context: A267486 A285229 A227425 * A301636 A238857 A253587

Adjacent sequences:  A333210 A333211 A333212 * A333214 A333215 A333216

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Mar 14 2020

STATUS

approved

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Last modified July 7 16:08 EDT 2020. Contains 335496 sequences. (Running on oeis4.)