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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)
OBJECTIVE This collaborative study was designed to collect data from multiple laboratories, using recognized abrasion and cleaning power methods, to assess the functionality, stability, and suitability of three Sident silica abrasives for consideration as references for ISO (International Standards Organization) abrasion and other dentifrice assessment(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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