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Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas. Following an introduction to the basic properties of Catalan numbers, the book presents 214 different kinds of objects counted by them in the form of exercises with solutions. The reader can try solving the exercises or simply browse through them. Some 68 additional exercises with prescribed difficulty levels present various properties of Catalan numbers and related numbers, such as Fuss-Catalan numbers, Motzkin numbers, Schröder numbers, Narayana numbers, super Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The book ends with a history of Catalan numbers by Igor Pak and a glossary of key terms. Whether your interest in mathematics is recreation or research, you will find plenty of fascinating and stimulating facts here.

Catalan numbers (Mathematics) --- Combinatorial analysis. --- Matrices.

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This Special Issue presents research papers on various topics within many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and includes the most recent advances made in the area of symmetric functions and polynomials.

generalized Laguerre --- central complete Bell numbers --- rational polynomials --- Changhee polynomials of type two --- Euler polynomials --- generalized Laguerre polynomials --- Hermite --- conjecture --- Legendre --- the degenerate gamma function --- trivariate Lucas polynomials --- perfectly matched layer --- third-order character --- Euler numbers --- two variable q-Berstein operator --- entropy production --- hypergeometric function --- q-Bernoulli numbers --- q-Bernoulli polynomials --- symmetry group --- Bernoulli polynomials --- Fibonacci polynomials --- central incomplete Bell polynomials --- Chebyshev polynomials --- convolution sums --- Lucas polynomials --- Jacobi --- the modified degenerate Laplace transform --- q-Volkenborn integral on ?p --- and fourth kinds --- two variable q-Berstein polynomial --- the modified degenerate gamma function --- two variable q-Bernstein operators --- reduction method --- identity --- elementary and combinatorial methods --- generalized Bernoulli polynomials and numbers attached to a Dirichlet character ? --- explicit relations --- recursive sequence --- Fubini polynomials --- p-adic integral on ?p --- generating functions --- q-Euler number --- acoustic wave equation --- congruence --- trivariate Fibonacci polynomials --- stochastic thermodynamics --- fermionic p-adic integrals --- Laguerre polynomials --- fluctuation theorem --- Bernoulli numbers and polynomials --- w-torsion Fubini polynomials --- non-equilibrium free energy --- hypergeometric functions 1F1 and 2F1 --- recursive formula --- Chebyshev polynomials of the first --- second --- central complete Bell polynomials --- Apostol-type Frobenius–Euler polynomials --- sums of finite products --- q-Euler polynomial --- symmetric identities --- stability --- fermionic p-adic q-integral on ?p --- Gegenbauer polynomials --- continued fraction --- thermodynamics of information --- well-posedness --- fermionic p-adic integral on ?p --- catalan numbers --- classical Gauss sums --- three-variable Hermite polynomials --- q-Changhee polynomials --- Catalan numbers --- two variable q-Bernstein polynomials --- q-Euler polynomials --- analytic method --- representation --- mutual information --- Fibonacci --- Legendre polynomials --- Gegenbauer --- generalized Bernoulli polynomials and numbers of arbitrary complex order --- Lucas --- elementary method --- new sequence --- third --- the degenerate Laplace transform --- computational formula --- operational connection --- sums of finite products of Chebyshev polynomials of the third and fourth kinds --- Changhee polynomials --- linear form in logarithms