Congruences for Catalan and Motzkin numbers and related sequences
Emeric Deutsch (Polytechnic U.), Bruce E. Sagan (Michigan State U.)
TL;DR
This paper establishes new congruences for Catalan, Motzkin, and related sequences using combinatorial and algebraic methods, confirming several conjectures and highlighting the Thue-Morse sequence's role.
Contribution
It introduces novel congruences for these sequences based on binomial coefficient expressions, resolving existing conjectures.
Findings
Proved new congruences for Catalan and Motzkin numbers.
Connected sequences to the Thue-Morse sequence in various contexts.
Settled multiple conjectures by Cloitre and Zumkeller.
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 Benoit Cloitre and Reinhard Zumkeller. The Thue-Morse sequence appears in several contexts.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Mathematical Dynamics and Fractals