The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) maximal independent sets and the number of unlabeled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus regarding maximal chord diagrams enumerated up to rotations.
Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are $\aleph_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classific…
A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We provide tight lower and upper bounds on the number of $\varepsilon$-near perfect matchings in generalized quasirandom graphs. Moreover, based on these results, we provide a deterministic polynomial time algorithm that for a given dense graph $G$ of order $n$ and a real number $\varepsilon>0$, returns either a conclusion that $G$ has no $\varepsilon$-near pe…
Answering several questions of Duffus, Frankl and Rödl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.
Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborh…
For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U ø ordered by the inclusion are characterized as: (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph H_n, for some n>2; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of μdisjoint complete graphs of size ν, where μν=ω.
A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic closure of F_q. In this note, we use generating functions to give elementary proofs of some of their results, and some extensions.
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).
In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed dimension over a given finite field. We show how this method works for the explicit example of $2$-dimensional algebras over the field $\mathbb{F}_{2}$.
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations given by a character polynomial, we compute $H^i(Y_n; W_n)$ for all $i$ and all $n$ in terms of double generating functions. One consequence of the computation is a new recurrence phenomenon: the stable twisted Betti numbers $\lim_{n\to\infty}\dim H^i(Y_n; W_n)$ are linearly recurren…
In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type \tilde{A}_n in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type \tilde{A}_n. As a by-product, this provides an alternative proof for the number of quivers of Dynkin type D_n which was first determined by Buan and Torkildsen.
Homomorphism counting for immersion-closed classes is not isomorphism
Universidad de Valparaíso（瓦尔帕莱索大学）；Millennium Nucleus for Social Data Science (SODAS)（社会数据科学千年核心）；University of Manitoba（曼尼托巴大学）；University of Alaska Fairbanks（阿拉斯加费尔班克斯大学）
We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^nn!/|W|$ for the number of maximal chains in the noncrossing partition lattice $NC(W)$ of a real reflection group $W$. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the $W$-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.
Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
Let $\mathbb F_q$ be the finite field with $q$ elements and, for each positive integer $n$, let $A_q(n)$ be the number of non isomorphic functional graphs arising from $\mathbb F_q$-linear maps $T:\mathbb F_{q}^n\to \mathbb F_{q}^n$. In 2013, Bach and Bridy proved that, if $q$ is fixed and $n$ is sufficiently large, the quantity $\frac{\log \log A_q(n)}{\log n}$ lies in the interval $[\frac{1}{2}, 1]$. By combining some ideas from linear algebra, combinatorics and number theory, in this paper we provide sharper estimates on the function $A_q(n)$ and, in particular, we prove that $\lim\limits_{…
In a former paper the authors counted the number of non-isomorphic Mathon maximal arcs of degree 8 in PG(2,2^h), h not 7 and prime. In this paper we will show that in PG(2,2^7) a special class of Mathon maximal arcs of degree 8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then count the total number of non-isomorphic Mathon maximal arcs of degree 8. Finally we show that the special arcs found in PG(2,2^7) extend to two infinite families of Mathon arcs of degree 8 in PG(2,2^k), k odd an…
The maximal number of totally mixed Nash equilibria in games of several players equals the number of block derangements, as proved by McKelvey and McLennan.On the other hand, counting the derangements is a well studied problem. The numbers are identified as linearization coefficients for Laguerre polynomials. MacMahon derived a generating function for them as an application of his master theorem. This article relates the algebraic, combinatorial and game-theoretic problems that were not connected before. New recurrence relations, hypergeometric formulas and asymptotics for the derangement coun…
Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $Γ$ of order $n$, the number of nowhere-zero $Γ$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $Γ$, let $ε_2(Γ)$ be the largest integer $d$ so that $Γ$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $Γ$-flows in $G$…
The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in which the maximal elements of $S$ and $T$ overlap. We investigate the number of good and all embeddings of a rooted poset $S$ in the family of all binary trees on $n$ elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset $S$ in the family of all planted plan…
By revisiting the Kirchhoff's Matrix-Tree Theorem, we give an exact formula for the number of spanning trees of a graph in terms of the quantum relative entropy between the maximally mixed state and another state specifically obtained from the graph. We use properties of the quantum relative entropy to prove tight bounds for the number of spanning trees in terms of basic parameters like degrees and number of vertices.
A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function $F(P_B)$ of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron $P_B$ associated to a building set $B$ is described as a morphism from the certain combinatorial Hopf algebra of building sets to quasisymmetric functions. We define the $q$-analog $F_q(P_B)$ and derive its determining recurrence relations. The $f$-polynomial of the nestohedron $P_B$ appears as the principal specialization of the quasisymmetric function $F_q(P_B)$.
Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $Γ(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.
Counting the number of maximal independent sets of graphs was started over $50$ years ago by Erdős and Mooser. The problem has been continuously studied with a number of variations. Interestingly, when the maximal condition of an independent set is removed, such the concept presents one of topological indices in molecular graphs, the so called Merrifield-Simmons index. In this paper, we applied the concept of bivariate generating function to establish the recurrence relations of the numbers of maximal independent sets of regualr $n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorph…
