Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6872100 | Discrete Applied Mathematics | 2015 | 10 Pages |
Abstract
Recently Mansour and Shattuck studied (k,a)-paths and gave formulas that related the total number of humps in all (k,a)-paths to the number of super (k,a)-paths. These results generalized earlier results of Regev on Dyck paths and Motzkin paths. Their proofs are based on generating functions and they asked for bijective proofs for their results. In this paper we first give bijective proofs of Mansour and Shattuck's results, then we extend our study to (n,m)-Dyck paths. We give a bijection that relates the total number of peaks in all (n,m)-Dyck paths to certain free (n,m)-paths when n and m are coprime. From this bijection we get the number of (n,m)-Dyck paths with exactly j peaks, which is a generalization of the well-known result that the number Dyck paths of order n with exactly j peaks is the Narayana number 1knâ1kâ1nkâ1.
Keywords
Related Topics
Physical Sciences and Engineering
Computer Science
Computational Theory and Mathematics
Authors
Rosena R.X. Du, Yingying Nie, Xuezhi Sun,