Realizable quadruples for hex-polygons. Combinatorics of honeycombs
Let \( H \) be the set of the vertices of the hexagons in a tiling of \( \mathbb R^{2} \) by regular hexagons with unit area. An \textit{\( H \)-polygon} is a simple polygon whose vertices are all in \( H \). \( H \)-polygons arise in the study of honeycombs, which were introduced by \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] to prove the saturation theorem for representations of the Lie group \( GL_{n}(\mathbb C) \). The article under review proves several results for \( H \)-polygons analogous to theorems of \textit{G. Pick} [Sitzungsber. Lotos Prag (2) 19, 311--319 (1900; JFM 31.0215.03)], \textit{E. Ehrhart} [C. R. Acad. Sci., Paris 241, 686--689 (1955; Zbl 0065.03106)], and \textit{P. R. Scott} [Bull. Aust. Math. Soc. 15, 395--399 (1976; Zbl 0333.52002)] for polygons with vertices in \( \mathbb Z^{2} \). The article also includes three open problems. Given an \( H \)-polygon, let \( v(P) \) be the number of its vertices, let \( b(P) \) be the number of \( H \)-points on its boundary, and let \( i(P) \) be the number of \( H \)-points in its interior. Define the \textit{boundary characteristic} \( c(P) \) of \( P \) as follows. For each \( x \in H \cap \partial P \), let \( c(x, P) \) be the number of edges of the hexagonal tiling that extend locally from \( x \) into the exterior of \( P \), minus the number of such edges that extend locally from \( x \) into the interior of \( P \). Let \( c(P) = \sum_{x \in H \cap \partial P} c(x, P) \). An \( H \)-polygon \( P \) is called \textit{mensurable} if its area \( A(P) \) satisfies the following Pick-type formula: \[ A(P) = \frac{b(P)}{4} + \frac{i(P)}{2} + \frac{c(P)}{12} - 1. \] Finally, let \( q(P) = (v(P), b(P), i(P), c(P)) \in \mathbb Z^{4} \). The paper is devoted to proving several inequalities and parity conditions satisfied by the subsets \( \mathcal Q \), \({\mathcal{Q}}_{\text{co}}\), and \({\mathcal{Q}}_{\text{m}}\) of \( \mathbb Z^{4} \), where \[ \begin{aligned}\mathcal Q & = \{q(P):P\text{ is an } H\text{-polygon}\}, \\ \mathcal{Q}_{\text{co}} & = \{q(P): P \text{ is a convex \(H\)-polygon}\}, \\ \mathcal{Q}_{\text{m}} & = \{q(P): P \text{ is a mensurable \(H\)-polygon}\}.\end{aligned} \] Examples of the results obtained include the following: If \( (v,b,i,c) \in \mathcal Q \), then \( 6 - b \leq c \leq 6 + b\) (Theorem 1). If \( (v,b,i,c) \in \mathcal Q \), then \( b \) and \( c \) are of the same parity (Theorem 3). \noindent If \( (3, b, 1, c) \in {\mathcal Q}_{\text{co}} \), then \( b \in \{3, 4, 5, 6, 7, 8, 10\} \) and \( b+c \) is an even number from the set \( \{6, 8, \dotsc, 6 + 2b\} \) (Theorem 4). \noindent We have \( (3,3,k,3) \in {\mathcal Q}_{\text{co}} \cap {\mathcal Q}_{\text{m}} \) if and only if \( k \) is positive and odd. We have \( (3,3,t,9) \in {\mathcal Q}_{\text{co}} \cap {\mathcal Q}_{\text{m}} \) if and only if \( t \) is non-negative and even (Theorem 5). \noindent If \( (v, b, i, c) \in {\mathcal Q}_{\text{co}} \cap {\mathcal Q}_{\text{m}} \) and \( i \geq 1 \), then \( b \leq 6i + c + 12 - 2v \) (Theorem 6). Given \( q = (v, b, i, c) \in \mathbb Z^{4} \) with \( v \geq 7 \) and \( i \leq 1 \), we have \( q \in \mathcal Q \) if and only if \( q \notin {\mathcal Q}_{\text{co}} \) (Theorem 7).
scientific article; zbMATH DE number 5146109