A supercongruence involving Delannoy numbers and Schröder numbers

2486 on Delannoy Numbers and Schröder Numbers

The nth Delannoy number and the nth Schröder number given by D n = n k=0 n k n + k k and S n = n k=0 n k n + k k 1 k + 1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that p−1 k=1 D k k 2 ≡ 2 −1 p E p−3 (mod p) and p−1 k=1 S k m k ≡ m 2 − 6m + 1 2m 1 − m 2 − 6m + 1 p (mod p), where (−) is the Legendre symbol, E 0 , E 1 , E 2 ,. .. are Euler n...

Why Delannoy Numbers ?

Why Delannoy numbers?

This article is not a research paper, but a little note on the history of combinatorics: we present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. © 20...

Generalized Small Schröder Numbers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from (0, 0) to (2n, 0) with the step set of {(k, k), (l,−l), (2r, 0) | k, l, r ∈ P}, where P is the set of positive integers, which never goes below the x-axis, and with no horizontal steps at level 0. We find a bijection between 5-colored Dyck path...

Delannoy numbers and Legendre polytopes

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpr...