We describe a new basis of the ring of quasi-symmetric coinvariants, which is stable by the natural reversal of the set of variables. The indexing set is the set of triangulations of a regular ...polygon, instead of the set of Dyck paths used for the known basis. On décrit une nouvelle base de l'algèbre des coinvariants quasi-symétriques, qui est stable par l'involution naturelle et indexée par les triangulations d'un polygone régulier.
Le problème du premier chiffre décimal Fuchs, Aimé; Letta, Giorgio
The Electronic journal of combinatorics,
02/1996, Volume:
3, Issue:
2
Journal Article
Peer reviewed
Pour une large classe de parties de N$^*$ (à laquelle appartient notamment l'ensemble qui intervient dans le problème du premier chiffre décimal) on démontre que les densités arithmétiques inférieure ...et supérieure, ainsi que la densité logarithmique, coïncident avec les "densités conditionnelles" correspondantes, obtenues en conditionnant par rapport à l'ensemble P des nombres premiers.
Soit $U$ un ensemble de couples de lettres. Foata et Zeilberger ont introduit les $U$-statistiques pour les mots quelconques. Dans cette note, on établit une condition nécessaire et suffisante pour ...que les deux définitions "maj$_U$" et "maj2$_U$", qu'on rencontre dans le cas classique, sont équivalentes. Il est remarquable que cette condition est exactement la même que celle qui a été trouvée pour l'équidistribution des deux statistiques "maj$_U$" et "inv$_U$".
G. Anderson a développé une méthode nouvelle pour calculer l'intégrale de Selberg. Nous montrons que cette méthode s'applique aussi pour calculer une généralisation de l'intégrale de Selberg étudiée ...par J. Kaneko. Le résultat s'exprime à l'aide des polynômes de Jacobi symétriques à plusieurs variables. La preuve utilise les opérateurs de montée et de descente qui leur sont associés.
Let $G$ be a group. The power graph of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose ...power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group $G$, the clique number of the power graph of $G$ is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph as the enhanced power graph. For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal.
Distance-Regular Graphs Van Dam, Edwin R.; Koolen, Jack H.; Tanaka, Hajime
The Electronic journal of combinatorics,
04/2016, Volume:
1000
Journal Article
Peer reviewed
Open access
This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in ...the area of distance-regular graphs since the monograph 'BCN' Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989 was written.
Many Faces of Symmetric Edge Polytopes D'Alì, Alessio; Delucchi, Emanuele; Michałek, Mateusz
The Electronic journal of combinatorics,
07/2022, Volume:
29, Issue:
3
Journal Article
Peer reviewed
Open access
Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics — ...where they are called adjacency polytopes — and to Kantorovich-Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic combinatorial methods to investigate invariants of the associated symmetric edge polytopes.
Eigenvalues of Cayley Graphs Liu, Xiaogang; Zhou, Sanming
The Electronic journal of combinatorics,
04/2022, Volume:
29, Issue:
2
Journal Article
Peer reviewed
Open access
We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.
A graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex is called a $k-$friendship graph and denoted by $F_k$. This paper determines the graphs of order ...$n$ that have the maximum (adjacency) spectral radius among all graphs containing no $F_k$, for $n$ sufficiently large.
We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new ...partition identities related to the principally specialized characters of certain level $2$ modules for the affine Lie algebra $A_9^{(2)}$. Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic sum-sides which we believe to be new. All of the new conjectures presented in this article are supported by a strong mathematical evidence.
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