0,6

A pyramid in a Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.

Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 0, 1, 0, 0, 1, 0, 0, 1, ...](periodic sequence 0,0,1) DELTA [1, 1, 0, 1, 1, 0, 1, 1, 0, ...](periodic sequence 1,1,0), where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 18 2006

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176).

G.f. = G = G(t, z) satisfies z(1-tz)G^2-(1+z-2tz)G+1-tz = 0.

Sum_{k, 0<=k<=n}T(n,k) = A000108(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 18 2006

T(4,3)=5 because the Dyck paths of semilength 4 having pyramid weight 3 are: (ud)u(ud)(ud)d, u(ud)(ud)d(ud), u(ud)(ud)(ud)d, u(ud)(uudd)d, and u(uudd)(ud)d [here u=(1,1), d=(1,-1) and the maximal pyramids, of total length 3, are shown between parentheses].

Triangle begins:

[1],

[0, 1],

[0, 0, 2],

[0, 0, 1, 4],

[0, 0, 1, 5, 8],

[0, 0, 1, 7, 18, 16],

[0, 0, 1, 9, 34, 56, 32],

[0, 0, 1, 11, 55, 138, 160, 64]

Adjacent sequences: A091863 A091864 A091865 this_sequence A091867 A091868 A091869

Sequence in context: A138157 A073429 A123634 this_sequence A111146 A109077 A137585

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004