A q-Lagrange inversion theorem due to A. M. Garsia is proved by means of two sign-reversing, weight-preserving involutions on Catalan trees.

Suppose π = π1π2 · · ·πn is a partition of size n, represented in its canonical sequential form. We show that the number of partitions of size n so represented having no two adjacent letters the same and avoiding a single pattern of length five is given by the Catalan number Cn−1 in six particular instances. In addition to supplying apparently new combinatorial structures counted by the Catalan...

The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we link avoidance of one of the patterns to the harmonic numbers, while for three other patterns we show their distributions on 132-avoiding permutations are given...

We present a bijective proof of Shapiro’s convolution formula involving Catalan numbers of even index. As a corollary, we give a new interpretation of the Catalan numbers.

We relate Catalan numbers and Catalan determinants to random matrix integrals and to moments of spin representations of odd orthogonal groups. §

The present paper analyzes the production of the Catalan /o/-/ɔ/ contrast by six groups of Catalan-Spanish bilinguals residing in the island of Majorca. A process of phonetic convergence between the two languages is currently taking place, with some Catalan features spreading to Spanish and vice versa (Simonet 2008). Based on previous L2speech research (e.g. Flege et al. 1995, 1999, Oyama 1976)...

This study examines the acoustic realization of the Catalan /a/-/əә/ alternation as a function of lexical stress together with the production of the Majorcan Catalan stressed /əә/ by 30 Spanish-dominant and 30 Catalan-dominant early Spanish-Catalan bilinguals in Majorca (Spain). The acoustic analyses show that there are no differences between language dominance groups in the implementation of t...

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

Substitution at codon Ser315 of katG gene, a reliable marker for isoniazid (INH) resistance and proliferate in spite of the loss of cell wall is called an L- form. Unstable L- forms  can convert to normal cells but stble L-forms can not.cell wall - deffective state may be induced spontaneously or through the action of an appropriate agent such as antibiotic interfering with the synthesis of cel...

Let p be any odd prime. We mainly show that p−1 ∑ k=1 2k k (3k k ) ≡ 0 (mod p)