Generalized parking function polytopes

A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of length $n$ is an $n$-dimensional polytope in $\mathbb{R}^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in (Amanbayeva and Wang 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-parking function polytopes. This allows us to answer a conjecture of (Behrend et al. 2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.Comment: 29 pages, 3 figures, comments welcome

Tiling Representations of Zeckendorf Decompositions

Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f1 = 1 and f2 = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf’s theorem

Umbeli Belli Rock Shelter, a forgotten piece from the puzzle of the Middle Stone Age in KwaZulu-Natal, South Africa

Lithic technology in the Middle Stone Age (MSA) of southern Africa is key to reconstructing human daily life, people's interaction with their environment and technological and cultural change through time. Ongoing discussions about the evolution of technology in the MSA debate the causes of lithic variability within and between different assemblages across southern Africa. The well-known MSA sites such as Blombos Cave, Klasies River, Diepkloof and Sibudu serve as anchors for comparative studies by providing high resolution stratigraphies that cover long parts of the archaeological sequence. Researchers, however, should recognize that these and other key sites are often situated many hundreds kilometers away from each other and are located in diverse geographic settings, which raises questions about their comparability. It is therefore important to consider the archaeological signatures from less spectacular sites to help identify regional patterns and test models of cultural change at smaller spatial scales. KwaZulu-Natal serves as an excellent starting point to bring questions about continuity and change within the MSA into clearer focus, since the province contains several sites in close proximity to each other in comparable environments. Many of these sites, however, are still understudied or have even been forgotten completely. In this paper we describe the archaeological sequence of one such site, Umbeli Belli near Scottburgh. This site was excavated in 1979 by Charles Cable and contributes important information to the regional record of the MSA in KwaZulu-Natal