arXiv subjects

Zachary Louis Vasseur

Publications and source records attributed to Zachary Louis Vasseur.

4 recordsLinked to original sources

arXiv2026

Linear Recurrences of Generalized Schreier Sets Revisited

For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\ |\{F\subset\{1, 2, \ldots, n\}\,:\, q\min F\ge p|F|\mbox{ and }n\in F\}|.$$ Using the Inclusion-Exclusion Principle, Beanland et al. proved the recurrence $$|\mathcal{S}^{p/q}_{n}|\ =\ \sum_{k=1}^q(-1)^{k+1}\binom{q}{k}|\mathcal{S}^{p/q}_{n-k}| + |\mathcal{S}^{p/q}_{n-(p+q)}|.$$ We show that $(|\mathcal{S}^{p/q}_n|)_{n=1}^\infty$ is a subsequence with terms taken periodically from Padovan-like sequences which satisfy simple recurrence relations. As an application, we obtain an alternative proof of the above linear recurrence. Furthermore, a similar result holds for the sequence $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$ that counts maximal $(p,q)$-Schreier sets. We end with a discussion of the relation between $(|\mathcal{S}^{p/q}_{n}|)_{n=1}^\infty$ and $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$.

math.CO↗

arXiv2025

Linear Recurrences from Counting Schreier-Type Multisets

A nonempty set $F$ is Schreier if $\min F\ge |F|$. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have been discovered. Building on these works, we prove a linear recurrence for the sequence that counts multisets $F$ with $\min F\ge p|F|$. In particular, if we let $$\mathcal{A}^{(s)}_{p, n}\ :=\ \{F\subset \{\underbrace{1, \ldots, 1}_{s}, \ldots, \underbrace{n-1, \ldots, n-1}_{s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ then $$|\mathcal{A}^{(s)}_{p, n}| = \sum_{i=0}^s|\mathcal{A}^{(s)}_{p, n-1-ip}|.$$ If we color $s$ copies of the same integer by different colors from $1$ to $s$, i.e., $\mathcal{B}^{(s)}_{p, n}:= $ $$\{F\subset \{1_{1}, \ldots, 1_{s}, \ldots, (n-1)_1, \ldots, (n-1)_{s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ then $$|\mathcal{B}^{(s)}_{p, n}| = \sum_{i=0}^s \binom{s}{i}| \mathcal{B}^{(s)}_{p, n-1-ip}|.$$ Lastly, we count Schreier sets that do not admit multiples of a given integer $u\ge 2$ and witness linear recurrences whose coefficients are drawn from the $u$th row of the Pascal triangle and have alternating signs, except possibly the last one.

math.CO↗

arXiv2024

Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients $2$, there must be a coefficient $0$. This paper characterizes all positive integers that do not have $F_{2k}$ ($k\ge 1$) in their decompositions. This continues the work of Kimberling, Carlitz et al., Dekking, and Griffiths, to name a few, who studied such a characterization for Zeckendorf decomposition.

math.GM↗

arXiv2024

Weighted Schreier-type Sets and the Fibonacci Sequence

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $ω_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } ω_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ First, we prove that $$a_{k,k+\ell} \ =\ 2F_{k+\ell},\mbox{ for all }\ell\geqslant 0\mbox{ and }k\geqslant \ell+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > ω_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where $ω_{2,3}(E) = \sum_{i\in E, i\neq 2, 3}1$.

math.CO↗