The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the cars to have different sizes and each takes up a number of adjacent parking spaces after a trailer $T$ parked on the first $z-1$ spots. A preference sequence in which all the cars are able to park is called a parking sequence. In this paper, we study increasing parking sequences and count them via bijections to lattice paths with right boundaries. Then we study two notions of invariance in parking sequences and present various characterizations and enumerative results.
For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all $n$ drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called \emph{prime} parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the tota…
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 $|\{t\in \mathbf{t}\mid t\le i\}|+|\{c\in \mathbf{c}\mid c\le i\}|\ge i$ 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 correspond…
An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots,a_n)$ of positive integers whose increasing rearrangement $b_1\leq\cdots\leq b_n$ satisfies $b_i\leq k+(i-1)r$. The case $r=k=1$ corresponds to ordinary parking functions. We develop numerous properties of $(r,k)$-parking functions. In particular, if $F_n^{(r,k)}$ denotes the Frobenius characteristic of the action of the symmetric group $\mathfrak{S}_n$ on the set of all $(r,k)$-parking functions of length $n$, then we find a combinatorial interpretation of the coefficients of the power series $\left( \su…
Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking functions. By representing parking functions as labeled digraphs, Hivert, Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on $\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous to the product and coproduct of the Hopf algebra NCSym of symmetric functions in noncommuting variables defined in terms of the power sum basis. In this paper, we view a parking function as a word. We shall construct a Hopf algebra PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogo…
We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and have a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp(1) is a rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator introduced in [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of [14]. It includes as special cas…
We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. The $G$-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied $G$-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and an edge deletion $G \setminus e$ ($e$ is a non-bridge). We obtain purely combinatorial consequences of Tutte short…
Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $δ_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $δ_I$ is non-decreasing as a function of $r$ and non-increasing as a function of $d$. For vanishing ideals over finite fields, we show that $δ_I$ is strictly decreasing as a function of $d$ until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, $1$-dimensional and their associated primes are generated by linear fo…
