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Pamela E. Harris

Publications and source records attributed to Pamela E. Harris.

At least 19 recordsLinked to original sources
arXiv2026

Statistics on $\ell$-interval parking functions

The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among our results, we enumerate $\ell$-interval parking functions with respect to statistics such as inversion, displacement, and major index. We show that $1$-interval parking functions with fixed displacement exhibit a cyclic sieving phenomenon. We give closed formulas for the number of $1$-interval parking functions with a fixed number of inversions. We prove that a well-known bijection of Foata preserves the set of $\ell$-interval parking functions exactly when $\ell\leq 2$ or $\ell\geq n-2$, which implies that the inversion and major index statistics are equidistributed in these cases.

math.CO↗
arXiv2026

Parking Cascades: From the Simplest Sequence to Motzkin and Catalan

We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.

math.CO↗
arXiv2026

The Defective Parking Space and Defective Kreweras Numbers

A defective $(m,n)$-parking function with defect $d$ is a parking function with $m$ cars attempting to park on a street with $n$ parking spots in which exactly $d$ cars fail to park. We establish a way to compute the defect of a defective $(m,n)$-parking function and show that the defect of a parking function is invariant under the action of $\mathfrak{S}_m$, the symmetric group on $[m]=\{1,2,\ldots,m\}$. We introduce the defective parking space ${\sf DPark}_{m,n}$ spanned by defective parking functions and describe its Frobenius characteristic as an $\mathfrak{S}_m$ representation graded by defect via coefficients $\mathrm{Krew}_{d,n}(λ)$ called defective Kreweras numbers. We provide a conjectured formula for $\mathrm{Krew}_{d,n}(λ)$ for sufficiently large $n$. We also show that the set of nondecreasing defective $(m,n)$-parking functions with defect $d$ are in bijection with the set of standard Young tableaux of shape $(n + d, m - d)$. This implies that the number of $\mathfrak{S}_m$-orbits of defective $(m,n)$-parking functions with defect $d$ is given by $\frac{n-m+2d+1}{n+d+1}\binom{m+n}{n+d}$. We also give a multinomial formula for the size of an $\mathfrak{S}_m$-orbit of a nondecreasing $(m,n)$-parking function with defect $d$. We conclude by using these results to give a new formula for the number of defective parking functions.

math.CO↗
arXiv2026

Lehmer Parking Functions and Their Outcomes

We introduce Lehmer parking functions and study their set of parking outcomes. Our main results establish that the number of outcomes of Lehmer parking functions of length $n$ is given by a Bell number, which is exactly the number of set partitions of an $n$ element set. We also show that the number of outcomes of weakly decreasing Lehmer parking functions is given by a Catalan number, which corresponds to a subset of set partitions on a set with $n$ elements referred to as non-intersecting set partitions.

math.CO↗
arXiv2026

Counting $\ell$-interval Fubini rankings through their parking outcome

Fubini rankings with $n$ competitors are $n$-tuples with entries in $[n]=\{1,2,3,\ldots, n\}$ that encode the conclusion of a race that allows ties. Since Fubini rankings are parking functions, we can study their parking outcomes, which are permutations encoding the final parking order of the cars using the Fubini ranking as a preference list. We establish that the number of Fubini rankings with $n$ competitors having a fixed parking outcome $π$ is given by $2^{n-k}$, where $k$ denotes the number of runs in $π$. We then use this formula to give a new proof for the number of Fubini rankings, which is given by the Fubini numbers. We also consider the set of $\ell$-interval Fubini rankings with $n$ competitors, which are Fubini rankings where at most $\ell+1$ competitors tie at any rank. We show that the number of $\ell$-interval Fubini rankings with $n$ competitors having a fixed parking outcome $π$ is given by a product of a power of two and a product of $\ell$-Pingala numbers, where these factors depend only on the lengths of the runs that make up the parking outcome $π$. The $1$-interval Fubini rankings are known as unit Fubini rankings, and we show that the number of unit Fubini rankings having a fixed parking outcome $π$ is given by a product of Fibonacci numbers indexed by the lengths of the runs in $π$. We use these results to give a formula for the number of $\ell$-interval Fubini rankings with $n$ competitors for all $\ell\in[n]$. We conclude with some directions for further study.

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arXiv2026

On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions

We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$

math.CO↗
arXiv2025

A Probabilistic Parking Process and Labeled IDLA

In 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a favorite object of study in enumerative combinatorics. In our work, we introduce and study a probabilistic variant of the classical parking protocol, which is closely related to Internal Diffusion Limited Aggregation, or IDLA, introduced in 1991 by Diaconis and Fulton [19]. In particular, we compute the stationary distribution of this process when initiated with a particular class of initial preferences, of which weakly increasing parking functions are a subset. Furthermore, we compute the expected time it takes for the protocol to complete assuming all of the cars park, and prove that, in some cases, the parking process is negatively correlated. In addition, we study statistics of uniformly random weakly increasing parking functions such as the distribution of the last entry, the probability that a specific set of cars is lucky, and the expected number of lucky cars.

math.PR↗
arXiv2025

Lucky Cars in Fubini Rankings and Unit Fubini Rankings

We study lucky cars in subsets of parking functions, called Fubini rankings and unit Fubini rankings. A Fubini ranking is a sequence of nonnegative integers that encodes a valid ranking of competitors, where ties are allowed. A car (or competitor) is said to be lucky if it is the first instance of that rank appearing in the sequence. We present combinatorial characterizations and enumeration formulas for lucky cars in both Fubini rankings and unit Fubini rankings, and establish connections between these objects and ordered set partitions, as well as integer compositions. To obtain our results, we use several techniques to enumerate statistics over these families of objects. In particular, we employ generating functions, bijective and combinatorial arguments, recurrence relations, and Zeilberger's creative telescoping method.

math.CO↗
arXiv2025

Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars

In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars.

math.CO↗
arXiv2025

Inversions in parking functions

In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval parking functions to give analogous results. We conclude by introducing a probabilistic approach through which we obtain formulas for the total number of inversions and several other statistics across all parking functions and other sets of words closed under rearrangement.

math.CO↗
arXiv2025

The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

For integral weights $λ$ and $μ$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $μ$ in the irreducible representation with highest weight $λ$, which we denote by $m(λ,μ)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(λ,μ)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(λ,μ)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tildeα,μ)$, where $\tildeα$ is the highest root and $μ$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tildeα,μ)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $μ=-(α_i+α_{i+1}+\cdots+α_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.

math.RT↗
arXiv2025

Partial permutohedra

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is the convex hull of all vectors in $\{0,1,\ldots,n\}^m$ whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of $\mathcal{P}(m,n)$, and our methods and results include the following. For any $m$ and $n$, we obtain a bijection between the nonempty faces of $\mathcal{P}(m,n)$ and certain chains of subsets of $\{1,\dots,m\}$, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the $h$-polynomial of $\mathcal{P}(m,n)$. For any $m$ and $n$ with $n\ge m-1$, we use a pyramidal subdivision of $\mathcal{P}(m,n)$ to establish a recursive formula for the normalized volume of $\mathcal{P}(m,n)$, from which we then obtain closed expressions for this volume. We also use a sculpting process (in which $\mathcal{P}(m,n)$ is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of $\mathcal{P}(m,n)$ with arbitrary $m$ and fixed $n\le 3$, the normalized volume of $\mathcal{P}(m,4)$ with arbitrary $m$, and the Ehrhart polynomial of $\mathcal{P}(m,n)$ with fixed $m\le4$ and arbitrary $n\ge m-1$.

math.CO↗
arXiv2025

Enumerating Flat Fubini Rankings

Recall that the set of Fubini rankings on $n$ competitors consists of the $n$-tuples that encode the possible rankings of $n$ competitors in a competition allowing ties. Moreover, recall that a run (weak run) in a tuple is a subsequence of consecutive ascents (weak ascents). If the leading terms of the set of maximally long runs (weak runs) of a tuple are in increasing (weakly increasing) order, then the tuple is said to be flattened (weakly flattened). We define the set of strictly flattened Fubini rankings, which is the subset of Fubini rankings with runs of strict ascents whose leading term are strictly increasing. Analogously, we define the set of weakly flattened Fubini rankings, which is the subset of Fubini rankings with runs of weak ascents whose leading terms are in weakly increasing order. Our main results give formulas for the enumeration of strictly flattened Fubini rankings and weakly flattened Fubini rankings. We also provide some conjectures for further study.

math.CO↗
arXiv2025

Pullback parking functions

We introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters $k$ and $\ell$, respectively. In this setting, there are $n$ spots on a one-way street and $m$ cars attempting to park in those spots, and $1\leq m\leq n$. We let $α=(a_1,a_2,\ldots,a_m)\in[n]^m$ denote the parking preferences for the cars, which enter the street sequentially. Car $i$ drives to their preference $a_i$ and parks there if the spot is available. Otherwise, car $i$ checks up to $k$ spots behind their preference, parking in the first available spot it encounters if any. If no spots are available, or the car reaches the start of the street, then the car returns to its preference and attempts to park in the first spot it encounters among spots $a_i+1,a_i+2,\ldots,a_i+\ell$. If car $i$ fails to park, then parking ceases. If all cars are able to park given the preferences in $α$, then $α$ is called a $(k,\ell)$-pullback $(m,n)$-parking function. Our main result establishes counts for these parking functions in two ways: counting them based on their final parking outcome (the order in which the cars park on the street), and via a recursive formula. Specializing $\ell=n-1$, our result gives a new formula for the number of $k$-Naples $(m,n)$-parking functions and further specializing $m=n$ recovers a formula for the number of $k$-Naples parking functions given by Christensen et al. The specialization of $k=\ell=1$, gives a formula for the number of vacillating $(m,n)$-parking functions, a generalization of vacillating parking functions studied by Fang et al., and the $m=n$ result answers a problem posed by the authors. We conclude with a few directions for further study.

math.CO↗
arXiv2025

Probabilistic $(m,n)$-Parking Functions

In this article, we establish new results on the probabilistic parking model (introduced by Durmíc, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in [0,1]$, we study the parking preference of the last car, denoted $a_m$, and determine the conditional distribution of $a_m$ and compute its expected value. We show that both formulas depict explicit dependence on the probability parameter $p$. We study the case where $m = cn $ for some $ 0 < c < 1 $ and investigate the asymptotic behavior and show that the presence of ``extra spots'' on the street significantly affects the rate at which the conditional distribution of $ a_m $ converges to the uniform distribution on $[n]$. Even for small $ \varepsilon = 1 - c $, an $ \varepsilon $-proportion of extra spots reduces the convergence rate from $ 1/\sqrt{n} $ to $ 1/n $ when $ p \neq 1/2 $. Additionally, we examine how the convergence rate depends on $c$, while keeping $n$ and $p$ fixed. We establish that as $c$ approaches zero, the total variation distance between the conditional distribution of $a_m$ and the uniform distribution on $[n]$ decreases at least linearly in $c$.

math.PR↗
arXiv2024

Parking functions with a fixed set of lucky cars

In a parking function, a lucky car is a car that parks in its preferred parking spot and the parking outcome is the permutation encoding the order in which the cars park on the street. We give a characterization for the set of parking outcomes arising from parking functions with a fixed set of lucky cars. This characterization involves the descent bottom set of a permutation, and we use the characterization to we give a formula for the number of parking functions with a fixed set of lucky cars. Our work includes the cases where the number of cars is equal to the number of parking spots, and where there are more spots than cars. We also give product formulas for the number of weakly increasing parking functions having a fixed set of lucky cars, and when the number of cars equals the number of spots this is a product of Catalan numbers.

math.CO↗
arXiv2024

A tree approach to the happy function

In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power $e$ and base $b$, we consider happy numbers with at most $k$ digits and we give a formula for the cardinality of the preimage of a single iteration of the happy function. We show that these happy numbers arise naturally as children of a given vertex in the tree. We conclude by applying this technique to $e$-power $b$-unhappy numbers of a given height.

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