Congruences for Catalan and Motzkin numbers and related sequences

Abstract

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas' congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…31. The kernel R n coincides with the two-sided ideal in the free algebra F n−1 generated by elements of the form…”

“…31. The Fulton universal ring R n−1 is defined to be the quotient 26 16) where in the summation we assume additionally that the sets [i a , i a +j a ] := {i a , i a +1, .…”

“…Let MS = {n ∈ N | n = 2 2k (2r + 1) − 1, k ≥ 1, r ≥ 0} be a subset of the set of all odd integers [31]. Show that (a) M S n ≡ 1 (mod 2), if either n ≡ 0 (mod 2) or n ∈ MS n , (b) M S n ≡ 0 (mod 2), if n is an odd integer and n / ∈ MS. Show that P n (0, α, β) α α=0 = N n−1 (β + 1), where as before, N n (t) denotes the Narayana polynomial.…”

Untitled

“…It is known that C n is odd if and only if n = 2 m − 1 for some m; see for instance [Deutsch and Sagan 2006]. Hence, C n being odd implies that…”

Catalan numbers, primes, and twin primes

“…6]; an online Addendum is continuously updated). Quite a lot is known about the divisibility of the Catalan numbers; see [2,10]. They are obviously closely related to the middle binomial numbers and not surprisingly, there is a considerable literature on their divisibility too; see [13,5,19,15,17,16].…”