Vincent Delecroix
Rauzy classes define a partition of the set of irreducible (or indecomposable) permutations. They were defined by G. Rauzy as part of an induction algorithm for interval exchange transformations. In this article we prove an explicit formula for the cardinality of all Rauzy classes.
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Multiple conventions have been adopted for denoting Interval Exchange Transformations (IETs). The "non-labeled" convention was the original, while the "labeled" convention has proven convenient when investigating Flat Surfaces as described by IETs. We establish the relationship between Extended Rauzy Classes, an equivalence class of combinatorial data related to IETs, under each convention.
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich in [KZ03] and correspond bijectively to the connected components of the strata of the moduli space of abelian differentials. In a paper [Boi14] Boissy proposed a Rauzy-type dynamics that acts on a s…
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action are related to components of the moduli spaces of Abelian differentials with prescribed singularities, and, in two variants of the problem, have been classified by Kontsevich and Zorich, and by Boissy, through methods involving both combinatorics and algebraic geom…
We bound the number of nearly orthogonal vectors with fixed VC-dimension over $\setpm^n$. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and the generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy function.
The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or general position number) of G. In the paper, we obtain an upper bound and a lower bound regarding gp-number in all cactus with k cycles and t pendant edges. Furthermore, the gp-number of wheel graph is determined.
For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).
A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $γ(G)$, is the cardinality of the smallest dominating set of $G$. The domination polynomial is defined by $D(G,x) = \sum d(G,i)x^i$ where $d(G,i)$ is the number of dominating sets in $G$ with cardinality $i$. Two graphs $G$ and $H$ are considered $\mathcal{D}$-equivalent if $D(G,x)=D(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}$-equivalent to $G$. Extending pre…
Let $P_n$ denote the undirected path of length $n-1$. The cardinality of the set of congruence classes induced by the graph homomorphisms from $P_n$ onto $P_k$ is determined. This settles an open problem of Michels and Knauer (Disc. Math., 309\ (2009)\ 5352-5359). Our result is based on a new proven formula of the number of homomorphisms between paths.
Let $n$ be a positive integer, $σ$ be an element of the symmetric group $\mathcal{S}_n$ and let $σ$ be a cycle of length $n$. The elements $α,β\in \mathcal{S}_n$ are $σ$-equivalent, if there are natural numbers $k$ and $l$, such that $σ^k α=βσ^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $α= σ^{k_1} βσ^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph $Γ_n$ with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalenc…
Super-strong (elsewhere referred to as strong) Wilf equivalence is a type of Wilf equivalence on words that was introduced by Kitaev et al. in 2009. We provide a necessary and sufficient condition for two permutations in $n$ letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in (Kitaev et al., 2009) that the cardinality of each Wilf equivalence class is a po…
A set partition $σ$ of $[n]=\{1,\cdots ,n\}$ contains another set partition $ω$ if a standardized restriction of $σ$ to a subset $S\subseteq[n]$ is equivalent to $ω$. Otherwise, $σ$ avoids $ω$. Sagan and Goyt have determined the cardinality of the avoidance classes for all sets of patterns on partitions of $[3]$. Additionally, there is a bijection between the set partitions and restricted growth functions (RGFs). Wachs and White defined four fundamental statistics on those RGFs. Sagan, Dahlberg, Dorward, Gerhard, Grubb, Purcell, and Reppuhn consider the distributions of these statistics over v…
On Tuza's conjecture in co-chain graphs
针对Tuza猜想在弦图及其子类区间图等图类上仍未解决的问题，本文证明该猜想对划分两侧顶点数均为偶数的共链图（区间图的一个子类）成立。
Let $G=(V,E)$ be a simple graph. For any integer $k\geq 1$, a subset of $V$ is called a $k$-tuple total dominating set of $G$ if every vertex in $V$ has at least $k$ neighbors in the set. The minimum cardinality of a minimal $k$-tuple total dominating set of $G$ is called the $k$-tuple total domination number of $G$. In this paper, we introduce the concept of upper $k$-tuple total domination number of $G$ as the maximum cardinality of a minimal $k$-tuple total dominating set of $G$, and study the problem of finding a minimal $k$-tuple total dominating set of maximum cardinality on several clas…
Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general position set if no element of $S$ lies on a geodesic between any two other elements of $S$. The cardinality of a largest general position set is the general position number ${\rm gp}(G)$ of $G.$ In \cite{ullas-2016} graphs $G$ of order $n$ with ${\rm gp}(G)$ $\in \{2, n, n-1\}$ were characterized. In this paper, we characterize the classes of all connected graph…
A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if every vertex $v\not\in S$ lies on a shortest path between two vertices of $S$. The minimum cardinality of a geodetic set of $G$ is the geodetic number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of $G$ such that every vertex of $G$ belongs to the set of vertices of a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order o…
In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections that interchange crossings and nestings. For types B and C, they generalize a construction by Kasraoui and Zeng for type A, whereas for type D, we were only able to construct a bijection between non-crossing and non-nesting set partitions. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of the maximal crossing with the cardinality of the maximal nesting, as given by Chen, Deng, D…
We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.
A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $γ_t(G)$. The total dominating set $D$ is called a total co-independent dominating set if $V(G)\setminus D$ is an independent set and has at least one vertex. The minimum cardinality of any total co-independent dominating set is denoted by $γ_{t,coi}(G)$. In this paper, we show that, for any tree $T$ of order $n$ and diameter at least three, $n-β(T)\leq …
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The e…
A set partition $σ$ of $[n]=\{1,\dots,n\}$ contains another set partition $π$ if restricting $σ$ to some $S\subseteq[n]$ and then standardizing the result gives $π$. Otherwise we say $σ$ avoids $π$. For all sets of patterns consisting of partitions of $[3]$, the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previo…
Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there is a w in U such that (w,v) is in E(D). Let beta(D) be the cardinality of the largest independent set of D whose vertices are from different partite classes of D. Our main result says that there exists a h=h(beta(D)) such that D has a dominating set of size at most h. This result is applied to settle a problem related to generalized Gallai colo…
We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to noncrossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the stable sets of maximal cardinality. We investigate how the class of $0/1$-polytopes whose edges satisfy our characterization is situated within the hierarchy of $0/1$-polytopes. This includes the cl…
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