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Register a new userEnumerating Parking Completions Using Join and Split
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Given a strictly increasing sequence $mathbf{t}$ with entries from $[n]:={1,ldots,n}$, a parking completion is a sequence $mathbf{c}$ with $|mathbf{t}|+|mathbf{c}|=n$ and $|{tin mathbf{t}mid tle i}|+|{cin mathbf{c}mid cle i}|ge i$ for all $i$ in $[n]$. We can think of $mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences $mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed textbf{Join} and textbf{Split}. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the signature parking functions of Ceballos and Gonzalez DLeon.
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This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitions. Using these bijections, we enumerate both ascending and descending $k$-Naples parking functions.
The subdivision graph $mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1dot{vee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1underline{vee}G_2$, is the graph obtained from $mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, where $I(G_1)$ is the set of inserted vertices of $mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs. We also give the number of the spanning trees and the Kirchhoff index of $G_1dot{vee}G_2$ (respectively, $G_1underline{vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the nth Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.
The author presents a computer implementation, calculating the terms of the Saneblidze-Umble diagonals on the permutahedron and the associahedron. The code is analyzed for correctness and presented in the paper, the source code of which simultaneously represents both the paper and the program.
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