A generalization of Catalan numbers

Abstract

In this paper, we introduce a generalization of Catalan numbers. To obtain this extension, we construct a family of subsets which depend on three parameters and whose cardinals originate it. The elements of this family are used to classify canonical primitive connected matrices of the p-Sylow of GLn(q), problem that is related to Higman’s Conjecture, which asserts that if Gn is the subgroup of GLn(q) consisting of upper unitriangular matrices, then the number of conjugacy classes of Gn is a polynomial in q. The construction of these subsets allows us to prove by elementary way the recurrence relations and properties of our generalization of Catalan numbers. The associated sequences of integers can be arranged in tables called s-triangles. If s=1, the 1-triangle is the Catalan triangle. Consequently, to particularize the identities and properties of the s-triangles to the 1-triangle, we can deduce identities of Catalan numbers already proved. Moreover, for s≤5 the first diagonals of the s-triangles are well-known sequences of integers which arise in many mathematical scopes.