"Enumerative result never comes for free, but only after one has elucidated, at least partly, what the objects really are. "

-- Mireille Bousquet-Mélou

"While counting, you notice that your knowledge about the subject is not sufficiently precise, and so counting becomes an educational activity. "

-- N.G. de Bruijn

"Combinatorics is the area of mathematics that is concerned with, relates to, employs, or studies combinatorial arguments. So what is a combinatorial argument? [..] A combinatorial argument is the one that consists predominately of ingenuity or detailed reasoning rather than knowledge of existing mathematics. This is in contrast with knowledge-based argument, which relies heavily on piecing together known results. "

-- D.Kleitman

"Combinatorics can be classified into three types: enumerative, existential, and constructive. Enumerative combinatorics deals with the counting of combinatorial objects. Existential combinatorics studies the existence or nonexistence of combinatorial configurations. Constructive combinatorics deals with methods for actually finding specific configurations (as opposed to merely demonstrating their existence theoretically). [..] In constructive combinatorics, the problem is usually one of finding a solution efficiently, [..] using a reasonable length of time. "

-- G. Pólya, R.E. Tarjan and D.R. Woods

"Once, Israil Gelfand said that mathematics has three parts: analysis, geometry, and combinatorics. "What is combinatorics?" the listeners asked. The answer was: "This is a science not yet created ..." "

-- I.M.Gelfand

"Though combinatorics has been successfully applied to many branches of mathematics these can not be compared neither in importance nor in depth to the applications of analysis in number theory or algebra to topology, but I hope that time and the ingenuity of the younger generation will change this. "

-- P.Erdos

"It has been said that combinatorics has too many theorems, matched with very few theories; Stanley's book belies this assertion. "

-- Gian-Carlo Rota

"One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the modern combinatorial techniques, including Topological methods, Algebraic methods, and Probabilistic methods, merely supply existence proofs and do not yield efficient (deterministic or randomized) algorithms for the corresponding problems. "

-- N.Alon

"Combinatorics is the nanotechnology of mathematics. "

-- Sara Billey