A333755 Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Apr 10 2020

Keywords

• nonn
• tabl

Comments

Except for a(1) = 0, the data is identical to A238130 shifted right once. However, in A238130, each row after the first ends with a zero, while here each row after the first starts with a zero.

Examples

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   4   1   0
   0   2  10   4   0   0
   0   4  12  14   2   0   0
   0   2  22  29  10   1   0   0
   0   4  26  56  36   6   0   0   0
   0   3  34 100  86  31   2   0   0   0
   0   4  44 148 200  99  16   1   0   0   0
   0   2  54 230 374 278  78   8   0   0   0   0
Row n = 6 counts the following compositions (empty column indicated by dot):
  .  (6)       (15)     (123)    (1212)
     (33)      (24)     (132)    (2121)
     (222)     (42)     (141)
     (111111)  (51)     (213)
               (114)    (231)
               (411)    (312)
               (1113)   (321)
               (1122)   (1131)
               (2211)   (1221)
               (3111)   (1311)
               (11112)  (2112)
               (21111)  (11121)
                        (11211)
                        (12111)
		

Crossrefs

Removing all zeros gives A238279.

The version for anti-runs is A106356.

The k-th composition in standard-order has A124767(k) runs.

The version counting descents is A238343.

The version counting weak ascents is A333213.

Cf. A066099, A124762, A238130, A272919, A333381, A333382, A333489.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k&]],{n,0,10},{k,0,n}]