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Vuong Bui

Publications and source records attributed to Vuong Bui.

At least 19 recordsLinked to original sources
arXiv2026

An optimal refinement-compatible bijection between singleton-free partitions and partitions without cyclic adjacencies

It is well known that the number of partitions of $[n]$ without singletons equals the number of partitions of $[n]$ in which no block contains two cyclically adjacent elements $i,i+1\pmod{n}$. Bernhart remarked that there might be no simple bijection between these two classes. Although Callan later constructed an algorithmic bijection proving the stronger equidistribution of singletons and adjacencies, his construction proceeds through multiple rounds of exchanges. Therefore, Bernhart's remark may still retain some validity, as suggested by Chen and Wang. In this article, we address this remark by giving a direct ``one-round'' bijection between the two classes. Unlike Callan's bijection, our map is closely compatible with the refinement order on partitions: in one direction it only decomposes blocks, while its inverse only merges blocks, with a single exceptional pair when $n>2$ is even. We further observe that this exception is unavoidable, establishing the optimality of the bijection with respect to the refinement order. The specific local form of these operations --- splitting off only singleton blocks and merging a singleton only with the block containing its cyclic neighbor --- also ensures that the construction restricts, without modification, to a bijection between the corresponding classes of noncrossing partitions.

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arXiv2026

A direct injection for the strong $q$-log-convexity of Touchard polynomials

We provide a direct injection for the well-known strong log-convexity of the Bell numbers $B_n$, that is $B_mB_n\le B_{m-1}B_{n+1}$ for every $1\le m\le n$. Our injection $\Pi_m\times\Pi_n\to \Pi_{m-1}\times\Pi_{n+1}$, where $\Pi_n$ denotes the set of all partitions of $[n]$, preserves the total number of blocks in the pair of partitions. In other words, it is also an injection for the strong $q$-log-convexity of Touchard polynomials, a result established by Chen, Wang, and Yang using analytical arguments. As an application of the injection, we also recover a related result of Chern, Diaconis, Kane, and Rhoades.

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arXiv2026

A sharper log-convexity inequality for Bell numbers

We prove a stronger version of the log-convexity inequality for the Bell numbers $B_n$. In particular, for $n\ge 5$, we have \[ B_{n+1}B_{n-1} - (B_n)^2 \ge \sum_{i=1}^{n} F_i (B_{n-i})^2, \] where $F_i$ is the $i$-th Fibonacci number with $F_0=F_1=1$. The simple proof is mostly combinatorial with elementary inequalities.

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arXiv2026

Explicit entropy bounds for symmetric nearest-neighbor subshifts

We provide another approach to Friedland's result that the topological entropy $h$ of a symmetric nearest-neighbor subshift is computable. Instead of the previous algebraic technique, our approach is mostly combinatorial and involves only counts of locally admissible patterns $C_n$ of a cube $[1,n]^d$ in $\mathbb Z^d$. The main idea is a reflection-gluing construction: we flip admissible patterns and merge them along their boundaries. In addition to a short and elementary proof, another advantage is that our approach yields an explicit convergence rate in arbitrary dimensions, whereas obtaining such a rate is already complicated for $\mathbb Z^3$ in Friedland's approach. In particular, we show that for every $n\ge 1$, \[ \frac{1}{n^d}(\log C_{n+1} - q_d(n)\log|\Sigma|) \le h \le \frac{1}{n^d} \log C_n, \] where $\Sigma$ is the alphabet and \[ q_d(n)=(2^d-1)\sum_{k=0}^{d-1} \frac{\binom{d}{k}}{2^d-2^k}\, n^k. \]

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arXiv2026

The mapping index through the lens of the cross-index

We study the cross-index of free \(G\)-posets as a combinatorial analogue of the equivariant topological index. We demonstrate that the cross-index exhibits many structural properties closely paralleling those of the topological index, while its behavior with respect to unions displays a pronounced dichotomy depending on the acting group. Specifically, if \(P = A \cup B\) is a union of \(G\)-invariant subposets, then for \(G = \mathbb{Z}_2\) we obtain the sharp inequality \[ \operatorname{xind} P \le \operatorname{xind} A + \operatorname{xind} B + 1, \] which is directly analogous to the classical union inequality for the topological index. In contrast, for every group \(G\neq \mathbb{Z}_2\), this phenomenon fails in general, and we establish the best possible weaker estimate \[ \operatorname{xind} P \le \operatorname{xind} A + 2(\operatorname{xind} B+1). \] This reveals a fundamental distinction between the \(\mathbb{Z}_2\)-equivariant and non-\(\mathbb{Z}_2\)-equivariant settings at the purely combinatorial level. As further consequences, we compare the cross-index with both the topological index and the simplicial index, showing in particular that the gap between the cross-index and the topological index can be arbitrarily large. These results clarify the role of the cross-index as a combinatorial analogue of the equivariant topological index and further strengthen the interplay between equivariant topological methods and combinatorial structures endowed with symmetry.

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arXiv2025

There exist infinite cube-free words over any sequence of binary alphabets

We prove that for any sequence of binary alphabets $\mathcal{A}_1,\mathcal{A}_2,\dots$, there exists a cube-free word $c_1c_2\dots$ so that $c_1\in\mathcal{A}_1,c_2\in\mathcal{A}_2,\dots$. In particular, for every $n$, there are at least $1.35^n$ cube-free words in $\mathcal{A}_1\times\mathcal{A}_2\times\dots\times \mathcal{A}_n$. We also prove that if the list of alphabets is computable then one of these words is computable and its $n$th letter can be computed in time polynomial in $n$.

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arXiv2025

A convolutional approach to bounding the number of polyominoes

Although known lower bounds for the growth rate $\lambda$ of polyominoes, or Klarner's constant, are already close to the empirically estimated value $4.06$, almost no conceptual progress on upper bounds has occurred since the seminal work of Klarner and Rivest (1973). Their approach, based on enumerating millions of local neighborhoods (also called ``twigs'') yielded $\lambda \le 4.649551$, later refined by Barequet and Shalah (2022) to $\lambda \le 4.5252$ using trillions of configurations. The inefficiency lies in representing each polyomino as an almost unrestricted sequence of neighborhoods once the large set of neighborhoods is fixed. We introduce a recurrence-based approach that constrains how local neighborhoods concatenate. Using a small system of convolution-type recurrences, we obtain $\lambda \le 4.5238$. The proof is short, self-contained, and hand-checkable. Despite the marginal numerical improvement, the main contribution is methodological: replacing trillions of configurations with a concise one-page system of recurrences. In addition, we present a new technique for rigorously bounding the growth of recurrences to any precision, applicable to a broad range of settings with nonnegative coefficients. The resulting upper bound even comes with a nice feature: a small set of parameters serves as the certificate for the bound, that is, one does not need to check more than a few arithmetic calculations to trust the bound.

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arXiv2025

A short proof of an upper bound on the growth constant of polyiamonds

We provide a short and elementary proof that the growth constant of polyiamonds is at most $1+2z+3z^2$ for the unique real root $z$ of the equation $2z^3+z^2-1=0$. This coincidentally suffices to recover the best known upper bound $3.6108$. Unlike the previous proof of this bound, which relied on computer-assisted technical arguments and the counts of polyiamonds with up to 75 triangles, our method is based on a straightforward recurrence that can be verified by hand with minimal effort.

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arXiv2025

A characterization of the Carath\'eodory number for $H$-convexity

We show that the Carath\'eodory number for $H$-convexity is the maximum of two parameters: the Helly number for $H$-convexity and the cone number of $H$. The cone number in this article is defined as the maximal number of points of $H$ in conical position with an empty positive hull relative to the remaining points. Earlier partial results by Boltyanski and Martini can provide an exact value for the Carath\'eodory number only when the Helly number is $1$ or $2$. We further establish connections between the Carath\'eodory numbers for $H$-convexity and that for $K$-strong convexity, where $H$ is the set of normals of $K$. Specifically, the Carath\'eodory number for $H$-convexity provides a lower bound for that of $K$-strong convexity. Moreover, if $K$ is a polytope, which has $|H|$ facets, then the Carath\'eodory number for $K$-strong convexity is at most the maximum of $|H|-1$ and the Carath\'eodory number for $H$-convexity. We conjecture a characterization of when the bound $|H|-1$ is attained. It is a consequence of a broader conjecture stating that the Carath\'eodory number for $K$-strong convexity is at most the maximum of the Carath\'eodory numbers for $H'$-convexity over all subsets $H'\subseteq H$. Finally, we observe that the Carath\'eodory number is at least the Helly number in any convex-structure where all sets are ordinarily convex.

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arXiv2024

Bounding Klarner's constant from above using a simple recurrence

Klarner and Rivest showed that the growth of the number of polyominoes, also known as Klarner's constant, is at most $2+2\sqrt{2}<4.83$ by viewing polyominoes as a sequence of twigs with appropriate weights given to each twig and studying the corresponding multivariate generating function. In this short note, we give a simpler proof by a recurrence on an upper bound. In particular, we show that the number of polyominoes with $n$ cells is at most $G(n)$ with $G(0)=G(1)=1$ and for $n\ge 2$, \[ G(n) = 2\sum_{m=1}^{n-1} G(m)G(n-1-m). \] It should be noted that $G(n)$ has multiple combinatorial interpretations in literature.

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arXiv2024

An explicit condition for boundedly supermultiplicative subshifts

We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient condition on $\mathcal{F}$ and $\mathcal{A}$ for the growth of $\mathcal{L}(\mathcal{A},\mathcal{F})$ to be boundedly supermultiplicative. That is, there exist constants $C>0$ and $\alpha\ge0$, such that for all $n$, the number of words of length $n$ in $\mathcal{L}(\mathcal{A},\mathcal{F})$ is between $\alpha^n$ and $C\alpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $\alpha$, the growth rate of the language, is also computable whenever our condition holds. We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $\mathcal{F}$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words.

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arXiv2024

Growth of recurrences with mixed multifold convolutions

Generalizing some popular sequences like Catalan's number, Schr\"oder's number, etc, we consider the sequence $s_n$ with $s_0=1$ and for $n\ge 1$, \begin{multline*} s_n=\sum_{x_1+\dots+x_{\ell_1}=n-1} \kappa_1 s_{x_1}\dots s_{x_{\ell_1}} + \dots +\sum_{x_1+\dots+x_{\ell_{t'}}=n-1} \kappa_{t'} s_{x_1}\dots s_{x_{\ell_{t'}}}+\\ \max_{x_1+\dots+x_{\ell_{t'+1}}=n-1} \kappa_{t'+1} s_{x_1}\dots s_{x_{\ell_{t'+1}}} + \dots + \max_{x_1+\dots+x_{\ell_t}=n-1} \kappa_t s_{x_1}\dots s_{x_{\ell_t}}, \end{multline*} where $x_i$ are nonnegative integers, $\ell_1,\dots,\ell_t$ are positive integers, and $\kappa_1,\dots,\kappa_t$ are positive reals. We show that it is possible to compute the growth rate $\lambda$ of $s_n$ to any precision. In particular, for every $n\ge 2$, \[ \sqrt[n]{\frac{\kappa^*}{\mathcal L(n-1) s_1} s_n} \le \lambda \le \sqrt[n]{3^{18\log 3 + 2\log\frac{s_1\mathcal L^2}{\kappa^*}} n^{3\log n + 12\log 3 + \log\frac{s_1\mathcal L^2}{\kappa^*}} s_n}, \]where $\mathcal L=\max_i \ell_i$ and $\kappa^*=\kappa_i$ for some $i$ with $\ell_i\ge 2$, and the logarithm has the base $\frac{\mathcal L+1}{\mathcal L}$. The constants in the inequalities are not very well optimized and serve mostly as a proof of concept with the ratio of the upper bound and the lower bound converging to $1$ as $n$ goes to infinity.

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arXiv2023

The number of polyiamonds is supermultiplicative

While the number of polyominoes is known to be supermultiplicative by a simple concatenation argument, it is still unknown whether the same applies to polyiamonds. This article proves that if $\ell,m$ are not both $1$, then $T(\ell+m)\ge T(\ell)T(m)$, for which one can say that the number of polyiamonds $T(n)$ is supermultiplicative. The method is, however, by concatenating, merging and adding cells at the same time. A corollary is an increment of the best known lower bound on the growth constant from $2.8423$ to $2.8578$.

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arXiv2023

A topological version of Hedetniemi's conjecture for equivariant spaces

A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for $G$-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of $G$-spaces. More precisely, we show that this conjecture can possibly survive if the group $G$ is either a cyclic $p$-group or a generalized quaternion group whose size is a power of 2.

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arXiv2022

An asymptotic lower bound on the number of polyominoes

Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ be Klarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We show that there exist some positive numbers $A,T$, so that for every $n$ \[ P(n) \ge An^{-T\log n} \lambda^n. \] This is somewhat a step toward the well known conjecture that there exist positive $C,\theta$ so that $P(n)\sim Cn^{-\theta}\lambda^n$ for every $n$. In fact, if we assume another popular conjecture that $P(n)/P(n-1)$ is increasing, we can get rid of $\log n$ to have \[ P(n)\ge An^{-T}\lambda^n. \] Beside the above theoretical result, we also conjecture that the ratio of the number of some class of polyominoes, namely inconstructible polyominoes, over $P(n)$ is decreasing, by observing this behavior for the available values. The conjecture opens a nice approach to bounding $\lambda$ from above, since if it is the case, we can conclude that \[ \lambda < 4.1141, \] which is quite close to the current best lower bound $\lambda > 4.0025$ and greatly improves the current best upper bound $\lambda < 4.5252$. The approach is merely analytically manipulating the known or likely properties of the function $P(n)$, instead of giving new insights of the structure of polyominoes. The techniques can be applied to other lattice animals and self-avoiding polygons of a given area with almost no change.

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arXiv2022

Every generating polytope is strongly monotypic

We prove an old conjecture of McMullen, Schneider and Shephard that every polytope with the generating property is strongly monotypic. The other direction is already known, which implies that strong monotypy and the generating property for polytopes are the same notion. A criterion for monotypic and strongly monotypic polytopes is also given.

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arXiv2022

Growth of Replacements

The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some $2$ balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some $n$, the object of the game is to find the maximum possible sum of values $g(n)$ for a table of $n$ balls. In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to converge to a growth rate $λ$. Furthermore, this value $λ$ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate $λ$ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given $λ_0$ is smaller than $λ$ is provided to approximate $λ$. When the values of the balls are all rational, we can compute the precise value of $λ$ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit $λ$ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements).

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