We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 4 2 , 4 3 , . . .) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k 2 , k 3 , . . .) for any k ≥ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t 2 + t, (t 2 + t) 2 , . . .).
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a oneto-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem on free Dyck paths and some generalizations and variations with respect to Dyck paths and Schröder paths with flaws turn out to be immediate consequences of the butterfly decomposition and the preorder traversal of plane trees. We obtain two involutions on free Dyck paths and free Schröder paths, leading to two combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of the generating function for the number of chains in plane trees due to Klazar. We further study the average size of chains in plane trees with n edges and show that this number asymptotically tends to n+9 6 .
Many interesting combinatorial objects are enumerated by the k-Catalan numbers, one possible generalization of the Catalan numbers. We will present a new combinatorial object that is enumerated by the k-Catalan numbers, staircase tilings. We give a bijection between staircase tilings and k-good paths, and between k-good paths and k-ary trees. In addition, we enumerate k-ary paths according to DD, UDU, and UU, and connect these statistics for k-ary paths to statistics for the staircase tilings. Using the given bijections, we enumerate statistics on the staircase tilings, and obtain connections with Catalan numbers for special values of k. The second part of the paper lists a sampling of other combinatorial structures that are enumerated by the k-Catalan numbers. Many of the proofs generalize from those for the Catalan structures that are being generalized, but we provide one proof that is not a straightforward generalization. We propose a web site repository for these structures, similar to those maintained by Richard Stanley for the Catalan numbers [R.P. Stanley, Catalan addendum. Available at: http://www-math.mit.edu/˜rstan/ec/] and by Robert Sulanke for the Delannoy numbers [R. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (1) (2003), Article 03, 1, 5, 19 pp. Available also at: math.boisestate.edu/˜sulanke/infowhowasdelannoy.html]. On the website, we list additional combinatorial objects, together with hints on how to show that they are indeed enumerated by the k-Catalan numbers.
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