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Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed$k$ -ary tree where we place$k^n$chips labeled$0,1,\\dots, k^n-1$on the root for some nonnegative integer$n$ , and we say a vertex$v$can fire if it has at least$k$chips. When a vertex fires, we select$k$labeled chips and send the$i$ th smallest chip among them to its$i$ th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of$1, 2, \\dots, n$ . We then express the stable configuration as a permutation of$0,1, 2, \\dots, k^n-1$and explore its properties, such as the number of inversions and descents. 20 pages, 4 figures, 1 table, v4: final version

One of the central goals of neuroscience is to understand how various animal behaviors are coordinated by the central nervous system. In many vertebrates, rhythmic outputs of neurons are known to be regulated by the neural networks called central pattern generators (CPGs) in the brainstem and the spinal cord. To understand the mechanism of CPG, the vocal behavior of African clawed frog, Xenopus laevis, has been used as a model system for more than a decade. Although numbers of research have been conducted to elucidate details of how populations of neurons in the vocal CPGs play crucial roles in generating Xenopus vocalization, the focus has been placed on the electrical and anatomical properties of neurons and neural circuitry, and less is known about molecular properties of neurons. To further explore the neural mechanisms underlying behavior, it is critical to identify neurochemicals that can activate or inhibit the specific neurons. With this premise, we have applied a technique called constellation pharmacology to the dorsal tegmental area of the medulla (DTAM), a region containing a component of the CPG to control male courtship calls in X. laevis. Using this method, various pharmacological agents were applied to the dissociated DTAM neurons while responses of each cells were simultaneously recorded by imaging calcium signal; based on the cell size and the response profiles to the applied pharmacological agents, every imaged cell was categorized into two major neural classes that can be further subdivided into multiple subclasses. One subset of dissociated DTAM neurons was sensitive to respond to both NMDA and GABA/Glycine and was considered as a putative fast trill neuron (FTN), a type of premotor neuron discovered to play a key role in generating a male courtship call. Further examination of these cells via constellation pharmacology revealed that some of these putative FTNs responded to substance P and acetylcholine.

Weighted Catalan numbers are a class of weighted sums over Dyck paths. Well-studied for their arithmetic properties and applications to enumerative combinatorics, these numbers were recently generalized to the setting of \\(k\\)-dimensional Catalan numbers for \\(k 2\\). In this paper, we introduce the \\(k\\)-dimensional semisymmetric weighted Catalan numbers (\\(k\\)-dimensional SSWCNs), an alternative \\(k\\)-dimensional generalization, along with their variant, the \\(k\\)-dimensional \\(u\\)-bounded semisymmetric weighted Catalan numbers (\\(k\\)-dimensional \\(u\\)-bounded SSWCNs). We define these two classes of numbers using the notion of semisymmetric height, a new statistic on points in \\(Z^k_ 0\\) motivated by geometric symmetries of \\(k\\)-dimensional analogs of Dyck paths and of the fundamental Weyl chamber of type \\(A_k-1\\). For our main results, we prove the eventual periodicity of \\(k\\)-dimensional SSWCNs and their \\(u\\)-bounded variants modulo a suitable integer \\(m\\), and we derive formulas for several classes of \\(k\\)-dimensional \\(u\\)-bounded SSWCNs. Additionally, using semisymmetric height, we derive novel analogs in the \\(k\\)-dimensional setting of the integer sequence counting Dyck paths by height and of the Narayana numbers. We conclude the paper with a future direction for generalizing weighted Catalan numbers to the \\(k\\)-dimensional setting.

We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of triangular arrays: the first one enumerates the \\(k\\)-dimensional Balanced ballot paths of exact height \\(s\\); the second one is a new multidimensional generalization of the Narayana numbers, which count the number of Balanced ballot paths with exactly \\(p\\) peaks.

We explore labeled chip-firing on undirected \\(k\\)-ary trees, trees where every vertex has degree \\(k+1\\). First, we extend known results for binary trees from Musiker and Nguyen, including the endgame and the locations of the smallest and largest chips, as well as relations between chips at different vertices. Then, inspired by recent work on the binary tree by the first author, Khovanova, and Luo, we use these properties to construct an upper bound, which we call the zigzag bound, on the number of stable configurations in labeled chip-firing on \\(k\\)-ary trees with \\(k^-1k-1\\) labeled chips starting at the root. We further provide a novel lower bound on the number of stable configurations of \\(k\\)-ary trees, complementing our upper bounds.

Chip-firing on a directed graph is a game in which chips, a discrete commodity, are placed on the vertices of the graph and are transferred between vertices. In this paper, we study a chip-firing game on the Hasse diagram of the lattice of nonnegative integer points on the plane, where we start with \\(2^n\\) chips at the origin. When we fire a vertex \\(v\\), we send one chip to each out-neighbor. We fire until we reach a stable configuration, a distribution of chips where no vertex can fire. We study the intermediate firing configuration: a table that assigns to each vertex the total number of chips that pass through it. We prove that the nonzero entries of the stable configuration correspond to the odd entries of the intermediate configuration. The intermediate configuration consists of three parts: the top triangle, the midsection, and the bottom triangle. We describe properties of each part. We study properties of each row and the number of rows of the intermediate configuration. We also explore properties of the difference tables, which are tables of first differences of each row of the intermediate firing configuration.

Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed \\(k\\)-ary tree where we place \\(k^n\\) chips labeled \\(0,1, k^n-1\\) on the root for some nonnegative integer \\(n\\), and we say a vertex \\(v\\) can fire if it has at least \\(k\\) chips. When a vertex fires, we select \\(k\\) labeled chips and send the \\(i\\)th smallest chip among them to its \\(i\\)th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of \\(1, 2, n\\). We then express the stable configuration as a permutation of \\(0,1, 2, k^n-1\\) and explore its properties, such as the number of inversions and descents.

Let \\(_d^(a,-)(n) = q_d^(a)(n) - Q_d^(a,-)(n)\\) where \\(q_d^(a)(n)\\) counts the number of partitions of \\(n\\) into parts with difference at least \\(d\\) and size at least \\(a\\), and \\(Q_d^(a,-)(n)\\) counts the number of partitions into parts \\( a d + 3\\) excluding the \\(d+3-a\\) part. Motivated by generalizing a conjecture of Kang and Park, Duncan, Khunger, Swisher, and the second author conjectured that \\(_d^(3,-)(n) 0\\) for all \\(d 1\\) and \\(n 1\\) and were able to prove this when \\(d 31\\) is divisible by \\(3\\). They were also able to conjecture an analog for higher values of \\(a\\) that the modified difference function \\(_d^(a,-,-)(n) = q_d^(a)(n) - Q_d^(a,-,-)(n) 0\\) where \\(Q_d^(a,-,-)(n)\\) counts the number of partitions into parts \\( a d + 3\\) excluding the \\(a\\) and \\(d+3-a\\) parts and proved it for infinitely many classes of \\(n\\) and \\(d\\). We prove that \\(_d^(3,-)(n) 0\\) for all but finitely many \\(d\\). We also provide a proof of the generalized conjecture for all but finitely many \\(d\\) for fixed \\(a\\) and strengthen the results of Duncan et.al. We provide a conditional proof of a linear lower bound on \\(d\\) for the generalized conjecture, which improves our unconditional result based on a conjectural modification of a recently proven conjecture of Alder. Using this modification, we obtain a strengthening of this generalization of Kang and Park's conjecture which remarkably allows \\(a\\) as a part. Additionally, we provide asymptotic evidence that this strengthened conjecture holds.

Chip-firing is a combinatorial game played on a graph, in which chips are placed and dispersed on the vertices until a stable configuration is achieved. We study a chip-firing variant on an infinite, rooted directed \\(k\\)-ary tree, where we place \\(k^n\\) chips labeled \\(1,2,3, k^n\\) on the root for some nonnegative integer \\(n\\). A vertex \\(v\\) can fire if it has at least \\(k\\) chips; when it fires, \\(k\\) chips are selected, and the chip with the \\(i\\)th smallest label is sent to the \\(i\\)th leftmost child of \\(v\\). A stable configuration is reached when no vertices can fire. In this paper, we prove numerous properties of the stable configuration, such as that chips land on vertices in ranges and the lengths of those ranges. We also describe where each chip can land. This helps us describe possible stable configurations of the game.

Chip-firing is a combinatorial game played on a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed \\(k\\)-ary tree, where we place \\(k^\\) chips on the root for some positive integer \\(\\), and we say a vertex \\(v\\) can fire if it has at least \\(k\\) chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than \\(k\\) chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable.