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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 one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the(More)
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(More)
A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of(More)
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,(More)
In this paper, we consider matchings avoiding partial patterns 1123 and 1132. We give a bijection between 1123-avoiding matchings with n edges and nonnegative lattice paths from (0, 2) to (2n, 0). As a consequence, the refined enumeration of 1123-avoiding matchings can be reduced to the enumeration of certain lattice paths. Another result of this paper is a(More)
Current optical-electronic-optical conversion (o-e-o) technology [1] requires propagating high-speed signals in the electronic domain and cascading separately packaged electronic and optoelectronic devices, resulting in increased cost, size, weight, power consumption, and heat radiation. For o-e-o conversion without the use of conventional electronics, we(More)
Leakage and dielectric breakdown of SiO<sub>2</sub> are studied for Cu interconnect structures with either stand-alone CoWP or two-layer CoWP+SiN caps. Without a post-CoWP plasma clean, there are many early fails and the dielectric breakdown exhibits bimodal behavior. By adding a plasma clean after CoWP deposition, the early fails can be eliminated and high(More)
This list is a compilation of combinatorial structures enumerated by the k-Catalan numbers C k n = 1 kn + 1 kn + 1 n = 1 (k − 1)n + 1 kn n for any positive integers k and n. We start by giving known combinatorial structures, together with references to where a proof can be found. The remainder of the list consists of new structures that are generalizations(More)
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