A333755 - OEIS

A333755

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

167

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

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

OFFSET

0,5

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.

LINKS

Table of n, a(n) for n=0..77.

EXAMPLE

Triangle begins:

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)

MATHEMATICA

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

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.

Sequence in context: A132339 A333941 A137676 * A238130 A238707 A181111

Adjacent sequences: A333752 A333753 A333754 * A333756 A333757 A333758

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Apr 10 2020

STATUS

approved