[Paper Review] Structure and asymptotics for Catalan numbers modulo primes using automata

Rob Burns|arXiv (Cornell University)|Jan 11, 2017

semigroups and automata theory12 references3 citations

TL;DR

This paper uses the automata method of Rowland and Yassawi to analyze Catalan numbers modulo primes, showing that the asymptotic density of indices n for which Cₙ ≡ 0 mod p is 1 for all primes p. It establishes that Cₙ ≡ −1 mod p when n = pᵏ − 1 and proves that p divides Cₙ whenever n ≡ d mod p for d in { (p+1)/2, ..., p−2 }, using state transitions in p-adic automata to derive structural and density results.

ABSTRACT

Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n: C_n \equiv 0 \mod p \}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv \{ \frac{p+1}{2}, \frac{p+3}{2}, ..., p-2 \} \mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.

Motivation & Objective

• To characterize the distribution of Catalan numbers modulo primes using finite automata.
• To determine the asymptotic density of indices n for which Cₙ ≡ 0 mod p.
• To identify structural patterns in the base-p digit expansions of n that determine divisibility of Cₙ by p.
• To extend known results on Catalan number divisibility to all primes p ≥ 5 using automata-based computation.

Proposed method

• Employing the automata method of Rowland and Yassawi to model Cₙ mod p via finite-state machines defined by polynomial transitions.
• Using the Cartier operator Λ_{d,d} to compute state transitions from polynomial representations of automaton states.
• Representing each state as a bivariate polynomial in x and y modulo p, with initial state R(x,y) = y(1 − 2xy − 2xy²).
• Computing transitions by evaluating Λ_{d,d}(s * Q^{p−1}) for each digit d ∈ {0, ..., p−1}, where Q(x,y) = x(y+1)² − 1.
• Analyzing state diagrams to identify loop states, zero states, and forbidden digit patterns in base-p expansions.
• Leveraging the linearity of the Cartier operator to deduce transitions for constant-value states from known base cases.

Experimental results

Research questions

• RQ1 What is the asymptotic density of the set {n : Cₙ ≡ 0 mod p} for any prime p?
• RQ2 For which base-p digit patterns in n does p divide Cₙ?
• RQ3 What is the value of Cₙ mod p when n = pᵏ − 1?
• RQ4 Can the automata framework be used to derive general divisibility rules for Catalan numbers modulo primes?

Key findings

• The asymptotic density of {n : Cₙ ≡ 0 mod p} is 1 for all primes p, meaning almost all Catalan numbers are divisible by p.
• If n ≡ d mod p for d ∈ {(p+1)/2, (p+3)/2, ..., p−2}, then p divides Cₙ.
• For n = pᵏ − 1, Cₙ ≡ −1 mod p, generalizing a known result for p = 2.
• The zero state in the automaton is a loop state, so once reached, the automaton remains in zero, implying Cₙ ≡ 0 mod p for all such n.
• The set of base-p numbers with no digits in {(p+1)/2, ..., p−2} has asymptotic density 0, explaining why Cₙ ≡ 0 mod p for almost all n.
• The automaton for Cₙ mod p contains at most p+3 states, with three fixed polynomials always present: y(1−2xy−2xy²), 2xy(y+1), and −(y+1).