Some Enumerations on Non-Decreasing Dyck Paths

Some Enumerations on Non-Decreasing Dyck Paths

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all nondecreasing Dyck paths of length 2n. We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.

Some strings in Dyck paths

On Generalized Dyck Paths

We generalize the elegant bijective proof of the Chung Feller theorem from the paper of Young-Ming Chen [The Chung-Feller theorem revisited, Disc. Math. 308 (2008), 1328–1329].

Counting Generalized Dyck Paths

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y...

Pattern - avoiding Dyck paths †

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P , we determine a formula for the number of Dyck paths covered by P , as well as for the ...