Hasil untuk "math.CO"

S. Fomin, A. Zelevinsky

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

C. U. Angeliya, T. Karthick, S. Huang

For a graph $G$, $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. In this paper, we show the following results: (i) If $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)\geq 3$, then $\chi(G)\leq \max\{6, \omega(G)\}$, and the bound is tight for each $\omega(G)\notin \{4,5\}$. (ii) If $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)= 4$, then $\chi(G)= 4$. These results extend the chromatic bounds known for the class of ($P_2+P_2$, $K_4-e$)-free graphs and for the class of ($P_2+P_3$, $K_4-e$)-free graphs, improve the bound of Chen and Zhang [arXiv:2412.14524 [math.CO], 2024] given for the class of ($P_2+P_4$, $K_4-e$)-free graphs, partially answer a question of Ju and the third author [Theor. Comp. Sci. 993 (2024) Article No.: 114465] on `near optimal colorable graphs', and a question of Schiermeyer (unpublished) on the chromatic bound for ($P_7$, $K_4-e$)-free graphs.

Hongyang Wang

In this paper, we establish an optimal $\chi$-binding function for $(P_2\cup P_4,\text{ diamond})$-free graphs. We prove that for any graph $G$ in this class, $\chi(G)\le 4$ when $\omega(G)=2$, $\chi(G)\le 6$ when $\omega(G)=3$, and $\chi(G)=\omega(G)$ when $\omega(G)\ge 4$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. This result extends the known chromatic bounds for $(P_2\cup P_3,\text{ diamond})$-free graphs by showing that $(P_2\cup P_4,\text{ diamond})$-free graphs admit the same $\chi$-binding function. It also refines the chromatic bound obtained by Angeliya, Karthick and Huang [arXiv:2501.02543v3 [math.CO], 2025] for $(P_2\cup P_4,\text{ diamond})$-free graphs.

Patrick A. Bennett, Amanda Priestley

An $r$-sunflower is a collection of $r$ sets such that the intersection of any two sets in the collection is identical. We analyze a random process which constructs a $w$-uniform $r$-sunflower free family starting with an empty family and at each step adding a set chosen uniformly at random from all choices that could be added without creating an $r$-sunflower with the previously chosen sets. To analyze this process, we extend results of the first author and Bohman arXiv:1308.3732v5 [math.CO], who analyzed a general random process which adds one object at a time chosen uniformly at random from all objects that can be added without creating certain forbidden subsets.

Mehrshad Taziki

The concept of the \textit{relative fractional packing number} between two graphs $G$ and $H$, initially introduced in arXiv:2307.06155 [math.CO], serves as an upper bound for the ratio of the zero-error Shannon capacity of these graphs. Defined as: \begin{equation*} \sup\limits_{W} \frac{\alpha(G \boxtimes W)}{\alpha(H \boxtimes W)} \end{equation*} where the supremum is computed over all arbitrary graphs and $\boxtimes$ denotes the strong product of graphs. This article delves into various critical theorems regarding the computation of this number. Specifically, we address its NP-hardness and the complexity of approximating it. Furthermore, we develop a conjecture for necessary and sufficient conditions for this number to be less than one. We also validate this conjecture for specific graph families. Additionally, we present miscellaneous concepts and introduce a generalized version of the independence number that gives insights that could significantly contribute to the study of the relative fractional packing number.

Toufik Mansour, Mark Shattuck

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.

V. Ivanov

Matteo Mucciconi, L. Petrov

Spin q-Whittaker symmetric polynomials labeled by partitions λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} were recently introduced by Borodin and Wheeler (Spin q-Whittaker Polynomials, 2017. arXiv preprint arXiv:1701.06292 [math.CO]) in the context of integrable sl2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {sl}_2$$\end{document} vertex models. They are a one-parameter deformation of the t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document} Macdonald polynomials. We present a new more convenient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, q-λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{-\lambda _1}$$\end{document} and qλN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{\lambda _N}$$\end{document} (where λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP. In a scaling limit as q↗1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\nearrow 1$$\end{document}, spin q-Whittaker polynomials turn into a new one-parameter deformation of the gln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {gl}_n$$\end{document} Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as q↗1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\nearrow 1$$\end{document} we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016. arXiv:1503.04117 [math.PR]), and relate it to spin Whittaker functions.

Mathilde Bouvel, Veronica Guerrini, Simone Rinaldi

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes (called slicings) which grow according to these succession rules. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.

James Haglund, Jeffrey B. Remmel, Andrew Timothy Wilson

We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

Sadia Noureen, A. Bhatti, Akbar Ali

The modified first Zagreb connection index $ZC_{1}^{*} $ZC1∗ is a graph invariant, initially appeared within a formula of the total electron energy of alternant hydrocarbons in 1972, and revisted in a recent paper [A. Ali, N. Trinajstić. A novel/old modification of the first Zagreb index. Mol Inform. 2018;37(6). Art# 1800008; arXiv:1705.10430 [math.CO]]. In this paper, the graph(s) with the maximum/minimum $ZC_{1}^{*} $ZC1∗ value is/are characterized from the class of all n-vertex trees with fixed number of segments. As the number of segments in a tree can be determined from the number of vertices of degree 2 (and vice versa), the trees with the extremum $ZC_{1}^{*} $ZC1∗ values are also determined from the class of all n-vertex trees having a fixed number of vertices of degree 2.

N. Fatima, A. Bhatti, Akbar Ali et al.

Abstract It is well known fact that several physicochemical properties of chemical compounds are closely related to their molecular structure. Mathematical chemistry provides a method to predict the aforementioned properties of compounds using topological indices. The Zagreb indices are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [Ali, A.; Trinajstić, N. A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO] 2017; Mol. Inform. 2018, 37, 1800008] and [Naji, A. M.; Soner, N. D.; Gutman, I. On leap Zagreb indices of graphs, Commun. Comb. Optim. 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we check the chemical applicability of the newly considered Zagreb connection indices on the set of octane isomers and establish general expressions for calculating these indices of two well-known dendrimer nanostars.

Zhenzhen Li, Baoyindureng Wu

A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

Saba Manzoor, N. Fatima, A. Bhatti et al.

Abstract The first Zagreb index (occurred in an approximate formula of total π-electron energy, communicated in 1972) and the second Zagreb index (appeared in 1975, within the study of molecular branching) are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [A. Ali, N. Trinajstić, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO], 2017] and [A. M. Naji, N. D. Soner, I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we derive formulas for calculating these modified versions of the Zagreb indices of four well known nanostructures.

Mehri Javanian

In a rooted tree, protected nodes are neither leaves nor parents of any leaves. They have some practical motivations, e.g., in organizational schemes, security models and social-network models. Protected node profile measures the number of protected nodes with the same distance from the root in rooted trees. For no rooted tree, protected node profile has been investigated so far. Here, we present the asymptotic expectations, variances, covariance and limiting bivariate distribution of protected node profile and non-protected internal node profile in random tries, an important data structure on words in computer science. Also we investigate the fraction of these expectations asymptotically. These results are derived by the methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization and depoissonization, saddle point method and singularity analysis.

John Campbell

To appear in Volume 19 of DMTCS.

S. Darbinyan

We prove two sharp sufficient conditions for hamiltonian cycles in balanced bipartite directed graph. Let $D$ be a strongly connected balanced bipartite directed graph of order $2a$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. (i) {\it If $a\geq 4$ and $max \{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then either $D$ is hamiltonian or $D$ is isomorphic to one exceptional digraph of order eight.} (ii) {\it If $a\geq 5$ and $d(x)+d(y)\geq 4a-3$ for every dominating pair of vertices $\{x,y\}$, then $D$ is hamiltonian.} The first result improves a theorem of R. Wang (arXiv:1506.07949 [math.CO]), the second result, in particular, establishes a conjecture due to Bang-Jensen, Gutin and Li (J. Graph Theory , 22(2), 1996) for strongly connected balanced bipartite digraphs of order at least ten.

A. Borodin, L. Petrov

Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.

Elizabeth Drellich

A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

G. Kuperberg

In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and 3-enumerations. Our main technical tool is a set of multi-parameter determinant and Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya determinant for UASMs [solv-int/9804010]. We evaluate specializations of the determinants and Pfaffians using the factor exhaustion method.