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Pamela E. Harris

Publications and source records attributed to Pamela E. Harris.

At least 73 records · Page 4Linked to original sources
arXiv2020

Enumerating Parking Completions Using Join and Split

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 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 González D'León.

math.CO↗
arXiv2020

Counting $k$-Naples parking functions through permutations and the $k$-Naples area statistic

We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before proceeding forward down the street. Note that the parking functions are the specialization of $k$ to $0$. For a fixed $0\leq k\leq n-1$, we define a function $φ_k$ which maps a $k$-Naples parking function to the permutation denoting the order in which its cars park. By enumerating the sizes of the fibers of the map $φ_k$ we give a new formula for the number of $k$-Naples parking functions as a sum over the permutations of length $n$. We remark that our formula for enumerating $k$-Naples parking functions is not recursive, in contrast to the previously known formula of Christensen et al [CHJ+20]. It can be expressed as the product of the lengths of particular subsequences of permutations, and its specialization to $k=0$ gives a new way to describe the number of parking functions of length $n$. We give a formula for the sizes of the fibers of the map $φ_0$, and we provide a recurrence relation for its corresponding logarithmic generating function. Furthermore, we relate the $q$-analog of our formula to a new statistic that we denote $\texttt{area}_k$ and call the $k$-Naples area statistic, the specialization of which to $k=0$ gives the $\texttt{area}$ statistic on parking functions.

math.CO↗
arXiv2020

Weight $q$-multiplicities for representations of the exceptional Lie algebra $\mathfrak{g}_2$

Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $ξ$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(ξ)=\sum c_i q^i$ where $c_i$ is the number of ways $ξ$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $\mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional Lie algebra $\mathfrak{g}_2$.

math.RT↗
arXiv2020

Generalized Lattice Point Visibility

It is a well-known result that the proportion of lattice points visible from the origin is given by $\frac{1}{ζ(2)}$, where $ζ(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika, generalized the notion of lattice point visibility by saying that for a fixed $b\in\mathbb{N}$, a lattice point $(r,s)\in\mathbb{N}^2$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f(x)=mx^b$, for some $m\in\mathbb{Q}$, between the origin and $(r,s)$. In their analysis they establish that for a fixed $b\in\mathbb{N}$, the proportion of $b$-visible lattice points is $\frac{1}{ζ(b+1)}$, which generalizes the result in the classical lattice point visibility setting. In this short note we give an $n$-dimensional notion of $\bf{b}$-visibility that recovers the one presented by Goins et. al. in $2$-dimensions, and the classical notion in $n$-dimensions. We prove that for a fixed ${\bf{b}}=(b_1,b_2,\ldots,b_n)\in\mathbb{N}^n$ the proportion of ${\bf{b}}$-visible lattice points is given by $\frac{1}{ζ(\sum_{i=1}^nb_i)}$. Moreover, we propose a $\bf{b}$-visibility notion for vectors $\bf{b}\in \mathbb{Q}_{>0}^n$, and we show that by imposing weak conditions on those vectors one obtains that the density of ${\bf{b}}=(\frac{b_1}{a_1},\frac{b_2}{a_2},\ldots,\frac{b_n}{a_n})\in\mathbb{Q}_{>0}^n$-visible points is $\frac{1}{ζ(\sum_{i=1}^nb_i)}$. Finally, we give a notion of visibility for vectors $\bf{b}\in (\mathbb{Q}^{*})^n$, compatible with the previous notion, that recovers the results of Harris and Omar for $b\in \mathbb{Q}^{*}$ in $2$-dimensions; and show that the proportion of $\bf{b}$-visible points in this case only depends on the negative entries of $\bf{b}$.

math.NT↗
arXiv2020

A formula for enumerating permutations with a fixed pinnacle set

In 2017 Davis, Nelson, Petersen, and Tenner pioneered the study of pinnacle sets of permutations and asked whether there exists a class of operations, which applied to a permutation in $\mathfrak{S}_n$, can produce any other permutation with the same pinnacle set and no others. In this paper, we adapt a group action defined by Foata and Strehl to provide a way to generate all permutations with a given pinnacle set. From this we give a closed non-recursive formula enumerating permutations with a given pinnacle set. Thus answering a question posed by Davis, Nelson, Petersen, and Tenner.

math.CO↗
arXiv2020

On the asymptotic behavior of the $q$-analog of Kostant's partition function

Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra $\mathfrak{g}$ as a sum of positive roots of $\mathfrak{g}$. We refer to each of these expressions as decompositions of a weight. Our main result considers an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the $q$-analog of Kostant's partition function and then prove that the (normalized) distribution of the number of positive roots in the decomposition of any of these weights converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We also extend these results to the highest root of the classical Lie algebras and we end our analysis with some directions for future research.

math.CO↗
arXiv2020

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$

The $q$-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostant's partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $\mathfrak{sl}_4(\mathbb{C})$ and give closed formulas for the $q$-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant's weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $\mathfrak{sl}_4(\mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.

math.CO↗
arXiv2019

Multi-color forcing in graphs

Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing procedure in terms of the number of vertices in the graph on which the procedure is being applied. We then focus on multi-color forcing with three colors and analyze the end states of certain families of graphs, including complete graphs, complete bipartite graphs, and paths, based on various initial colorings. We end with a few directions for future research.

math.CO↗
arXiv2019

Sequences of consecutive factoradic happy numbers

Given a positive integer $n$, the factorial base representation of $n$ is given by $n=\sum_{i=1}^ka_i\cdot i!$, where $a_k\neq 0$ and $0\leq a_i\leq i$ for all $1\leq i\leq k$. For $e\geq 1$, we define $S_{e,!}:\mathbb{Z}_{\geq0}\to\mathbb{Z}_{\geq0}$ by $S_{e,!}(0) = 0$ and $S_{e,!}(n)=\sum_{i=0}^{n}a_i^e$, for $n \neq 0$. For $\ell\geq 0$, we let $S_{e,!}^\ell(n)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}^0(n)=n$. If $p\in\mathbb{Z}^+$ satisfies $S_{e,!}(p)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in \mathbb{Z}^+$, if $p$ is an $e$-power factoradic fixed point and if there exists $\ell\in \mathbb{Z}_{\geq 0}$ such that $S_{e,!}^\ell(x)=p$, then we say that $x$ is an $e$-power factoradic $p$-happy number. Note an integer $n$ is said to be an $e$-power factoradic happy number if it is an $e$-power factoradic $1$-happy number. In this paper, we prove that all positive integers are $1$-power factoradic happy and, for $2\leq e\leq 4$, we prove the existence of arbitrarily long sequences of $e$-power factoradic $p$-happy numbers. A curious result establishes that for any $e\geq 2$ the $e$-power factoradic fixed points of $S_{e,!}$ that are greater than $1$, always appear in sets of consecutive pairs. Our last contribution, provides the smallest sequences of $m$ consecutive $e$-power factoradic happy numbers for $2\leq e\leq 5$, for some values of $m$.

math.NT↗
arXiv2019

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights λand μof the Lie algebras considered.

math.CO↗
arXiv2019

A Generalization of Parking Functions Allowing Backward Movement

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $n$-tuple containing the cars' parking preferences a parking function. In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $k$ spaces west of their preferred spot to park before proceeding east if all of those $k$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $k$-Naples parking functions of length $n$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $n$-tuples of positive integers $1$ to $n$, the case when $k\geq n-1$. Our main result provides a recursive formula for counting $k$-Naples parking functions of length $n$. We also give a characterization for the $k=1$ case by introducing a new function that maps $1$-Naples parking functions to classical parking functions, i.e. $0$-Naples parking functions. Lastly, we present a bijection between $k$-Naples parking functions of length $n$ whose entries are in weakly decreasing order and a family of signature Dyck paths.

math.CO↗
arXiv2019

When is the $q$-multiplicity of a weight a power of $q$?

Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights $(λ,μ)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram isomorphism) for which the multiplicity of the weight $μ$ in the representation of $\mathfrak{g}$ with highest weight $λ$ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated $q$-multiplicity for the pairs of weights considered, and conjecture that in all cases the $q$-multiplicity of such pairs of weights is given by a power of $q$.

math.CO↗
arXiv2019

A combinatorial model for computing volumes of flow polytopes

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes that were seemingly unapproachable. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.

math.CO↗
arXiv2019

Diversity of Artists in Major U.S. Museums

The U.S. art museum sector is grappling with diversity. While previous work has investigated the demographic diversity of museum staffs and visitors, the diversity of artists in their collections has remained unreported. We conduct the first large-scale study of artist diversity in museums. By scraping the public online catalogs of 18 major U.S. museums, deploying a sample of 10,000 artist records comprising over 9,000 unique artists to crowdsourcing, and analyzing 45,000 responses, we infer artist genders, ethnicities, geographic origins, and birth decades. Our results are threefold. First, we provide estimates of gender and ethnic diversity at each museum, and overall, we find that 85% of artists are white and 87% are men. Second, we identify museums that are outliers, having significantly higher or lower representation of certain demographic groups than the rest of the pool. Third, we find that the relationship between museum collection mission and artist diversity is weak, suggesting that a museum wishing to increase diversity might do so without changing its emphases on specific time periods and regions. Our methodology can be used to broadly and efficiently assess diversity in other fields.

stat.AP↗
arXiv2018

On the genus of a quotient of a numerical semigroup

We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ has embedding dimension $2$. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.

math.NT↗
arXiv2018

Bin Decompositions

It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any $n,m\in\mathbb{N}$ we create a sequence called the $(n,m)$-bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the $(n,m)$-bin legal decompositions displays Gaussian behavior.

math.CO↗
arXiv2018

Broadcast Domination of Triangular Matchstick Graphs and the Triangular Lattice

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer parameters $t$ and $r$. In this setting, a vertex $v \in V$ is a broadcast vertex of transmission strength $t$ if it transmits a signal of strength $t-d(u,v)$ to every vertex $u \in V$, where $d(u,v)$ denotes the distance between vertices $u$ and $v$ and $d(u,v) <t$. Given a set of broadcast vertices $S\subseteq V$, the reception at vertex $u$ is the sum of the transmissions from the broadcast vertices in $S$. The set $S \subseteq V$ is called a $(t,r)$ broadcast dominating set if every vertex $u \in V$ has a reception strength $r(u) \geq r$ and for a finite graph $G$ the cardinality of a smallest broadcast dominating set is called the $(t,r)$ broadcast domination number of $G$. In this paper, we consider the infinite triangular grid graph and define efficient $(t,r)$ broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient $(t,r)$ broadcasts on the infinite triangular lattice for all $t\geq r\geq 1$. Using these broadcasts, we then provide upper bounds for the $(t,r)$ broadcast domination numbers for triangular matchstick graphs when $(t,r)\in\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)\}$.

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