Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $\mu$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tilde{\alpha})$, where $\tilde{\alpha}$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in $L(\tilde{\alpha})$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tilde{\alpha})$ is precisely $q^{r-h(\mu)}$, where $h(\mu)=j-i+1$ is the height of the positive root $\mu$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one.
Singular cubical homology theory can be constructed for different categories of digraphs and quivers based on two types of digraph cubes. Furthermore, there exist various categories of digraphs and quivers that share the same set of objects but have different sets of morphisms. Additionally, there are different definitions of homotopy, leading to the existence of several homotopy categories in graph theory. Similarly, singular simplicial homology theories can be defined on various categories (homotopy categories) of digraphs and quivers. This paper systematically explores the relationships between singular homology theories for different categories of digraphs and quivers. We also introduce new notions of homology for these categories and describe the functorial and homotopy properties of them.
We show that with high probability (w.h.p.) the list chromatic number chi & ell; of the square of G(n,p )for p = c/n is asymptotically equal to the maximum degree Delta(G(n,p)). Since chi(G(n,p)(2)) < chi & ell;(G(2) (n,p)), this also improves an earlier result of Garapaty et al. [6] who proved that chi(G(n,p)(2)) < 6 & centerdot; Delta(G(n,p)) w.h.p. AMS 2000 SUBJECT CLASSIFICATIONS: Primary 05C80, 05C15.
In one of his papers on the weak order of Coxeter groups, Dyer formulates several conjectures. Among these, one affirms that the extended weak order forms a lattice, while another offers an algebraic-geometric description of the join of two elements in this poset. The former was recently proven for affine types by Barkley and Speyer. In this paper, we establish the latter for Coxeter groups of types A and I. Moreover, we verified the validity of this conjecture for types H_3 and F_4 through the use of Sage.
We study Nurikabe puzzles on non-orientable surfaces. Specifically, we propose two versions of non-orientable Nurikabe and investigate their combinatorics on Mobius strips, Klein bottles, and projective planes of size 1 & times; n. Our results establish new connections among the OEIS sequences A101946, A213387, A123203, and A001045 (the Jacobsthal sequence).
Recently, the general problem of enumerating permutations pi = pi(1)& centerdot;& centerdot;& centerdot; pi(n) such that pi(i+r)-pi(i) =/ s for all 1 <= i <= n-r, where r and s are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analogous problem on k-ary words involving the distribution of the corresponding statistic. Note that for k-ary words, it suffices to consider only the r = 1 case of the aforementioned problem on permutations. Here, we compute for arbitrary s an explicit formula for the ordinary generating function for n >= 0 of the distribution of the statistic on k-ary words rho = rho(1) & centerdot;& centerdot;& centerdot; rho n recording the number of indices i such that rho(i)+1-rho(i) = s. This result may then be used to find a comparable formula for finite set partitions with a fixed number of blocks, represented sequentially as restricted growth functions. Further, several sequences from the OEIS arise as enumerators of certain classes of k-ary words avoiding adjacencies with a prescribed difference. The comparable problem where one tracks indices i such that the absolute difference |a(i+1)-a(i)|is a fixed number is also considered on k-ary words and the corresponding generating function may be expressed in terms of Chebyshev polynomials.
Schur's Theorem states that, for any r is an element of & Zopf;(+), there exists a minimum integer S(r) such that every r-coloring of {1, 2, ... , S(r)} admits a monochromatic solution to x + y = z. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer GS(r) such that every r-coloring of {1, 2, ... , GS(r)} admits either a rainbow or monochromatic solution to x + y = z. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when x =/ y, we consider x + y + b = z and x + y < z, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to x + y = z and x + y < z.
We provide a framework for which one can approach showing the integer decomposition property for symmetric polytopes. We utilize this framework to prove a special case which we refer to as 2-partition maximal polytopes in the case where it lies in a hyper-plane of & Ropf;(3). Our method involves proving a special collection of polynomials that have saturated Newton polytope.
We find a generating function for interval-closed sets of the product of two chains poset by constructing a bijection to certain bicolored Motzkin paths. We also find a functional equation for the generating function of interval-closed sets of truncated rectangle posets, including the type A root poset, by constructing a bijection to certain quarter-plane walks.
The monotone path Pn+2 is an ordered 3-uniform hypergraph whose vertex set has size n + 2 and edge set consists of all consecutive triples. In this note, we consider the collection J(n) of ordered 3-uniform hypergraphs named monotone paths with n jumps, and we prove the following relation r (3; n) < R(Pn+2, Jn) < 4(n) center dot r (3; n), where r (3; n) is the multicolor Ramsey number for triangles and R(Pn+2, Jn) is the hypergraph Ramsey number for Pn+2 versus any member of Jn. In particular, whether r (3; n) grows exponentially, which is a very old problem of Erd & odblac;s, is equivalent to whether R(Pn+2, J(n)) grows exponentially.
We study a refinement of the q, t-Catalan numbers introduced by Xin and Zhang (2023, 2025) using tools from polyhedral geometry. These refined q, t-Catalan numbers depend on a vector of parameters (k) over right arrow and the classical q, t-Catalan numbers are recovered when (k) over right arrow = (1, ... , 1). We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on (k) over right arrow -Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on q, t-symmetry of the refined q, t-Catalan numbers in the cases where (k) over right arrow = (k(1), k(2), k(3)) and (k, k, k, k), give some extensions, including the case (k) over right arrow = (k, k+m, k+m, k+m), and discuss relationships to other generalizations of the q, t-Catalan numbers.
Inspired by the work of Amdeberhan, Can, and Moll on broken necklaces, we define a broken bracelet as a linear arrangement of marked and unmarked vertices and introduce a generalization called n-stars, which is a collection of n broken bracelets whose final (unmarked) vertices are identified. Through these combinatorial objects, we provide a new framework for the study of Kostant's partition function, which counts the number of ways to express a vector as a nonnegative integer linear combination of the positive roots of a Lie algebra. Our main result establishes that (up to reflection) the number of broken bracelets with a fixed number of unmarked vertices with nonconsecutive marked vertices gives an upper bound for the value of Kostant's partition function for multiples of the highest root of a Lie algebra of type A. We connect this work to multiplex juggling sequences, as studied by Benedetti, Hanusa, Harris, Morales, and Simpson, by providing a correspondence to an equivalence relation on n-stars.
We present a simplified variant of Biane's bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic socalled depth of a permutation. This generalizes a result by Guay-Paquet and Petersen about a continued fraction of the generating function for depth on the symmetric group Can n of permutations. In terms of weighted Motzkin path, we establish an involution on Can n that reverses the parities of depth and excedance numbers simultaneously, which proves that the numbers of permutations with even and odd depth (excedance numbers, respectively) are equal if n is even and differ by the tangent number if n is odd. Moreover, we present some interesting sign-imbalance results on permutations and derangements, refined with respect to depth and excedance numbers.
The dual immaculate and Young quasisymmetric Schur bases of quasisymmetric functions possess analogues in the peak algebra: respectively, the quasisymmetric Schur Q-functions and the peak Young quasisymmetric Schur functions. We show elements of the former basis expand into the latter basis with nonnegative coefficients.
A connected graph, on four or more vertices, is matching covered if every edge is present in some perfect matching. An ear decomposition theorem (similar to the one for $2$-connected graphs) exists for bipartite matching covered graphs due to Hetyei. From the results and proofs of Lov\'asz and Plummer, that rely on Hetyei's theorem, one may deduce that any minimal bipartite matching covered graph has at least $2(m-n+2)$ vertices of degree two (where minimal means that deleting any edge results in a graph that is not matching covered); such a graph is said to be extremal if it attains the stated lower bound. In this paper, we provide a complete characterization of the class of extremal minimal bipartite matching covered graphs. In particular, we prove that every such graph $G$ is obtained from two copies of a tree devoid of degree two vertices, say $T$ and $T'$, by adding edges -- each of which joins a leaf of $T$ with the corresponding leaf of $T'$. Apart from the aforementioned bound, there are four other bounds that appear in, or may be deduced from, the work of Lov\'asz and Plummer. Each of these bounds leads to a notion of extremality. In this paper, we obtain a complete characterization of all of these extremal classes and also establish relationships between them. Two of our characterizations are in the same spirit as the one stated above. For the remaining two extremal classes, we reduce each of them to one of the already characterized extremal classes using standard matching theoretic operations.
We study the Localization game on locally finite graphs trees, where each of the countably many vertices have finite degree. In contrast to the finite case, we construct a locally finite tree with localization number n for any choice of positive integer n. Our examples have uncountably many ends, and we show that this is necessary by proving that locally finite trees with finitely or countably many ends have localization number at most 2. Finally, as is the case for finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber.
The paper concerns the coarse flag Hilbert-Poincareseries of Maglione and Voll in the case of the braid arrangement B-n associated with the symmetric group & Sfr;(n+1. )We explicitly construct a companion statistic ino : & Sfr;(n+1) & times; Sym(n)-+ & Nopf; for the descent statistic on Sym(n) using reverse (P, w)-partitions and quasisymmetric functions.
For any integers 1 < k < n, we introduce a new family of parking functions called k-vacillating parking functions of length n. The parking rule for k-vacillating parking functions allows a car with preference p to park in the first available spot in encounters among the parking spots numbered p, p-k, and p+k (in that order and if those spots exist). In this way, k-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of & ell;-interval parking functions, which allow a car to park in its preference or up to & ell; spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the nth convergent of the continued fraction of \/2, as the number of non-decreasing 1-vacillating parking functions of length n. Our main result gives a product formula for the enumeration of k-vacillating parking functions of length n based on the number of 1-vacillating parking functions of smaller length. We conclude with some directions for further research.
The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime?
Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root mu in the adjoint representation of sl(r+1)(C), which we denote L((alpha) over tilde), where (alpha) over tilde is the highest root of sl(r+1)(C). We prove that the number of terms contributing a nonzero value to the multiplicity of the positive root mu = alpha(i) + alpha(i+1) + ... + alpha(j) with 1 <= i <= j <= r in L(alpha) is given by the product F-i . Fr-j+1, where F-n is the n(th) Fibonacci number. Using this result, we show that the q-multiplicity of the positive root mu = alpha(i) + alpha(i+1) + ... + alpha(j), with 1 <= i <= j <= r in the representation L(alpha) is precisely q(r-h(u)), where h(mu) = j - i + 1 is the height of the positive root mu. Setting q = 1 recovers the known result that the multiplicity of a positive root in the adjoint representation of sl(r+1)(C) is one.
