On the Congruences of Some Combinatorial Numbers


Address for correspondence: Shu-Chung Liu, Institute of Mathematics, Academia Sinica, Nankang, Taipei, 11529, Taiwan; e-mail: liularry@ms.ckitc.edu.tw


In this paper, we apply Lucas' theorem to evaluate the congruences of several combinatorial numbers, including the central Delannoy numbers and a class of Apéry-like numbers, the numbers of noncrossing connected graphs, the numbers of total edges of all noncrossing connected graphs on n vertices, etc. One of these results verifies a conjecture given by Deutsch and Sagan recently. In the end, we use an automaton to explain the idea of our approach.