We study a refinement of the \(q,t\)-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined \(q,t\)-Catalan numbers depend on a vector of ...parameters \(\vec{k}\) and the classical \(q,t\)-Catalan numbers are recovered when \(\vec{k} = (1,\ldots,1)\). We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on \(\vec{k}\)-Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on \(q,t\)-symmetry of the refined \(q,t\)-Catalan numbers in the cases where \(\vec{k} = (k_1,k_2,k_3)\) and \((k,k,k,k)\), give some extensions, including the case \(\vec{k} = (k,k+m,k+m,k+m)\), and discuss relationships to other generalizations of the \(q,t\)-Catalan numbers.
The Ehrhart polynomial \(\text{ehr}_P(n)\) of a lattice polytope \(P\) counts the number of integer points in the \(n\)-th integral dilate of \(P\). The \(f^*\)-vector of \(P\), introduced by Felix ...Breuer in 2012, is the vector of coefficients of \(\text{ehr}_P(n)\) with respect to the binomial coefficient basis \( \left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}\), where \(d = \dim P\). Similarly to \(h/h^*\)-vectors, the \(f^*\)-vector of \(P\) coincides with the \(f\)-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of \(f^*\)-vectors of polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of \(f\)-vectors of simplicial polytopes; e.g., the first half of the \(f^*\)-coefficients increases and the last quarter decreases. Even though \(f^*\)-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart \(h^*\)-vector, there is a polytope with the same \(h^*\)-vector whose \(f^*\)-vector is unimodal.
Let \(n\) points be in crescent configurations in \(\mathbb{R}^d\) if they lie in general position in \(\mathbb{R}^d\) and determine \(n-1\) distinct distances, such that for every \(1 \leq i \leq ...n-1\) there is a distance that occurs exactly \(i\) times. Since Erdős' conjecture in 1989 on the existence of \(N\) sufficiently large such that no crescent configurations exist on \(N\) or more points, he, Pomerance, and Palásti have given constructions for \(n\) up to \(8\) but nothing is yet known for \(n \geq 9\). Most recently, Burt et. al. had proven that a crescent configuration on \(n\) points exists in \(\mathbb{R}^{n-2}\) for \(n \geq 3\). In this paper, we study the classification of these configurations on \(4\) and \(5\) points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.
There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a ...poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable M\"obius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.
Res. number theory (2018) 4: 43 Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + \dots + c_t
H_{n-t}$ where $c_i \in \mathbb{N}_0$ for all $i$ and $c_1, c_t \geq 1$, the
generalized ...Zeckendorf decomposition (gzd) of $m \in \mathbb{N}_0$ is the
unique representation of $m$ using $H$ composed of blocks lexicographically
less than $\sigma = (c_1, \dots, c_t)$. We prove that the gzd of $m$ uses the
fewest number of summands among all representations of $m$ using $H$, for all
$m$, if and only if $\sigma$ is weakly decreasing. We develop an algorithm for
moving from any representation of $m$ to the gzd, the analysis of which proves
that $\sigma$ weakly decreasing implies summand minimality. We prove that the
gzds of numbers of the form $v_0 H_n + \dots + v_\ell H_{n-\ell}$ converge in a
suitable sense as $n \to \infty$, furthermore we classify three distinct
behaviors for this convergence. We use this result, together with the
irreducibility of certain families of polynomials, to exhibit a representation
with fewer summands than the gzd if $\sigma$ is not weakly decreasing.
We introduce a new family of \(N\times N\) random real symmetric matrix ensembles, the \(k\)-checkerboard matrices, whose limiting spectral measure has two components which can be determined ...explicitly. All but \(k\) eigenvalues are in the bulk, and their behavior, appropriately normalized, converges to the semi-circle as \(N\to\infty\); the remaining \(k\) are tightly constrained near \(N/k\) and their distribution converges to the \(k \times k\) hollow GOE ensemble (this is the density arising by modifying the GOE ensemble by forcing all entries on the main diagonal to be zero). Similar results hold for complex and quaternionic analogues. We isolate the two regimes by using matrix perturbation results and a nonstandard weight function for the eigenvalues, then derive their limiting distributions using a modification of the method of moments and analysis of the resulting combinatorics.
Zeckendorf proved that every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers. This has been extended in many ways, including to linear recurrences \(H_n=c_1 ...H_{n-1} + \cdots + c_t H_{n-t}\) where the \(c_i\) are non-negative integers and \(c_1\), \(c_t \ge 1\). Every integer has a unique generalized Zeckendorf decomposition (gzd) -- a representation composed of blocks that are lexicographically less than \((c_1,\dots,c_t)\), which we call the signature. We prove that the gzd of a natural number \(m\) uses the fewest number of summands out of all representations for \(m\) using the same recurrence sequence, for all \(m\), if and only if the signature of the linear recurrence is weakly decreasing (i.e., \(c_1 \ge \cdots \ge c_t\)). Following the parallel with well-known base \(d\) representations, we develop a framework for naturally moving between representations of the same number using a linear recurrence, which we then utilize to construct an algorithm to turn any representation of an integer into the gzd. To prove sufficiency, we show that if the signature is weakly decreasing then our algorithm results in fewer summands. To prove necessity we proceed by divide and conquer, breaking the analysis into several cases. When \(c_1 > 1\), we give an example of a non-gzd representation of an integer and show that it has fewer summands than the gzd by performing the same above-mentioned algorithm. When \(c_1 = 1\), we non-constructively prove the existence of a counterexample by utilizing the irreducibility of a certain family of polynomials together with growth rate arguments.
