[Paper Review] Asymptotic density of Motzkin numbers modulo small primes

Rob Burns|arXiv (Cornell University)|Nov 15, 2016

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper establishes the asymptotic density of Motzkin numbers modulo small primes (2, 3, 4, 5, 8) using structural characterizations of their residues. By analyzing sets defined by base-q expansions and applying asymptotic density theorems, it derives exact densities: 1/3 for even Motzkin numbers, 1/10 for those divisible by 5, and 1 for those divisible by 3.

ABSTRACT

We establish the asymptotic density of the Motzkin numbers modulo small primes.

Motivation & Objective

• To determine the asymptotic density of Motzkin numbers modulo small primes and prime powers, particularly 2, 3, 4, 5, and 8.
• To extend prior work on Motzkin number residues by quantifying the natural density of each residue class.
• To resolve open questions about the frequency of zero residues and other congruence classes in the Motzkin sequence.
• To provide a systematic method for computing asymptotic densities based on recursive residue structures and base-q expansions.

Proposed method

• Uses known structural characterizations of Motzkin number residues modulo small primes and powers of primes from prior work.
• Applies Theorem 1 to compute asymptotic densities of sets of the form $ S(q,r,s,t) = \{ (qi + r)q^{sj + t} \} $, where $ q, s > 0 $, $ t \geq 0 $, $ 0 \leq r < q $.
• Derives the asymptotic density of such sets as $ (q^{t+1-s}(q^s - 1))^{-1} $, enabling density computation for residue classes.
• Applies the density formula to specific forms of $ n $ for which $ M_n \equiv r \mod m $, based on prior results from Deutsch, Sagan, Eu, Liu, Yeh, Krattenthaler, Müller, and Rowland-Yassawi.
• Uses the fact that the set of numbers with only digits 0 and 1 in base 3 has asymptotic density zero to show that $ M_n \equiv 0 \mod 3 $ almost always.
• Combines disjoint residue classes and applies density additivity to compute total densities for each residue modulo 2, 3, 4, 5, and 8.

Experimental results

Research questions

• RQ1 What is the asymptotic density of Motzkin numbers congruent to 0 modulo 2?
• RQ2 What is the asymptotic density of Motzkin numbers congruent to 0 modulo 3?
• RQ3 What are the asymptotic densities of Motzkin numbers modulo 4, 5, and 8, particularly for each residue class?
• RQ4 How do the densities of Motzkin numbers modulo small primes compare to those of Catalan numbers?
• RQ5 Can the asymptotic density of Motzkin numbers modulo small primes be computed using structural forms derived from base-q expansions?

Key findings

• The asymptotic density of Motzkin numbers divisible by 2 is $ \frac{1}{3} $, derived from four disjoint residue classes each of density $ \frac{1}{12} $.
• The asymptotic density of Motzkin numbers congruent to 4 modulo 8 is $ \frac{1}{6} $, corresponding to two of the four residue classes.
• The asymptotic density of Motzkin numbers congruent to 2 or 6 modulo 8 is $ \frac{1}{12} $ each, due to balanced distribution of 1-bits in binary expansions.
• The asymptotic density of Motzkin numbers divisible by 5 is $ \frac{1}{10} $, with contributions from four disjoint forms: two of density $ \frac{1}{24} $ and two of density $ \frac{1}{120} $.
• The asymptotic density of Motzkin numbers divisible by 3 is 1, since the set of indices for which $ M_n \not\equiv 0 \mod 3 $ has density zero.
• The set of indices for which $ M_n \equiv 0 \mod 8 $ has asymptotic density zero, consistent with Rowland and Yassawi’s result that 0 is a forbidden residue modulo 8.