Jeffrey Shallit

collaborators

6 papers

math.NT2026

Balanced Fibonacci word rectangles, and beyond

Jeffrey Shallit, Ingrid Vukusic

Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved wi…

math.CO2026

On a sequence of Kimberling and its relationship to the Tribonacci word

LubomÃra DvoÅÃ¡kovÃ¡, Edita PelantovÃ¡, Jeffrey Shallit

In 2017, Clark Kimberling defined an interesting sequence $B = 0100101100 \dots$ of $0$ 's and $1$ 's by certain inflation rules, and he made a number of conjectures about thi…

cs.FL2026

State Complexity of Shifts of the Fibonacci Word

Delaram Moradi, Pierre Popoli, Jeffrey Shallit +1

The Fibonacci infinite word $f = (f_{i})_{i \geq 0} = 01001010 \dots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$ -state automaton tha…

math.NT2026

New properties of the $I ¨ $ -representation of integers

Jeffrey Shallit, Ingrid Vukusic

We prove a few new properties of the $I ¨ $ -representation of integers, where $I ¨ = (1 + 5) /2$ . In particular, we prove a 2012 conjecture of Kimberling. As software assistants, w…

cs.FL2025

The Tribonacci constant and finite automata

Jeffrey Shallit

We show that there is no automaton accepting the Tribonacci representations of $n$ and $x$ in parallel, where $I ¨ = 1.839 \dots$ is the Tribonacci constant, and $x= \lfloor n Ï\rfl…

math.NT2025

On the $p$ -adic valuations of values of Legendre polynomials

Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit +1

We prove an explicit formula for the $p$ -adic valuation of the Legendre polynomials $P_{n} (x)$ evaluated at a prime $p$ , and generalize an old conjecture of the third author. We also…