In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set $\{1,...,2n\}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$.
In this paper we derive polynomial time algorithms that generate random $k$-noncrossing matchings and $k$-noncrossing RNA structures with uniform probability. Our approach employs the bijection between $k$-noncrossing matchings and oscillating tableaux and the $P$-recursiveness of the cardinalities of $k$-noncrossing matchings. The main idea is to consider the tableaux sequences as paths of stochastic processes over shapes and to derive their transition probabilities.
Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a pair of noncrossing matchings satisfying one additional constraint. We prove that every noncrossing matching arises from some harmonic curve, and deduce from this that every basketball arises from some …
In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) \sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.
Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-…
We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of k-distant crossings and nestings is symmetric. We also study the numbers of k-distant noncrossing matchings and partitions for small k, which are counted by well-known sequences, as well as the orthogonal polynomials related to k-distant noncrossing matchings and par…
We give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of Chen, Deng, Du, Stanley, and Yan, and extends results of Klazar to k>2. Moreover, this correspondence re…
We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NP-completeness of deciding whether there is such a perfect matching. For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge. We also construct polygons with maximal compatible matc…
In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for $k$-noncrossing RNA structures. Our results are based on the generating function for the number of $k$-noncrossing RNA pseudoknot structures, ${\sf S}_k(n)$, derived in \cite{Reidys:07pseu}, where $k-1$ denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function $\sum_{n\ge 0}{\sf S}_k(n)z^n$ and obtain for $k=2$ and $k=3$ the analy…
A k-noncrossing RNA pseudoknot structure is a graph over $\{1,...,n\}$ without 1-arcs, i.e. arcs of the form (i,i+1) and in which there exists no k-set of mutually intersecting arcs. In particular, RNA secondary structures are 2-noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3-noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of \cite{Reidys:07rna1} and \cite{Reidys:07rna2}, where the generating function of k-noncrossing RNA pseudoknot structures and the asymptotics for…
In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${\sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $\ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $\sum_{n\ge 0}{\sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $k\le 9$ the asymptotic formula ${\sf T}_{k}^{[4]}(n)\sim c_k n^{-((k-1)^2+(k-1)/2)} γ_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $γ_k^{-1}$ and asymptotic formulas for $4\le k\le 9$.
The cyclic sieving phenomenon (CSP) was introduced by Reiner, Stanton, and White to study combinatorial structures with actions of cyclic groups. The crucial step is to find a polynomial, for example a q-analog, that satisfies the CSP conditions for an action. This polynomial will give us a lot of information about the symmetry and structure of the set under the action. In this paper, we study the cyclic sieving phenomenon of the cyclic group $C_{2n}$ acting on $P_{n,k}$, which is the set of matchings of $2n$ points on a circle with $k$ crossings. The noncrossing matchings ($k=0$) was recently…
A connected planar cubic graph is called an $m$-barrel fullerene and denoted by $F(m,k)$, if it has the following structure: The first circle is an $m$-gon. Then $m$-gon is bounded by $m$ pentagons. After that we have additional k layers of hexagons. At the last circle $m$-pentagons connected to the second $m$-gon. In this paper we asymptotically count by two different methods the number of perfect matchings in $m$-barrel fullerene graphs, as the number of hexagonal layers is large, and show that the results are equal.
We give upper bounds for the number $Φ_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular, for $d$-regular, $N$-vertex graphs, our bound is best possible up to an error factor of the form $\exp[o_d(1)N]$, where $o_d(1) \rightarrow 0$ as $d \rightarrow \infty$. This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markström. Some further possibilities are also suggested.
In this paper we review the asymptotic matching conjectures for $r$-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in $d$-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of $r$-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density $p$, for any $p\in [0,1]$, for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for ce…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that 'every edge coloured graph of a certain type contains a rainbow matching using every colour'. In this paper we introduce a versatile 'sampling trick', which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures. - We give a simple proof of Pokrovskiy's asymptotic v…
We study (asymmetric) $U$-statistics based on a stationary sequence of $m$-dependent variables; moreover, we consider constrained $U$-statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and…
We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to ques…
We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size $\ell$ in the graph consisting of $\frac{N}{2d}$ disjoint copies of the complete bipartite graph $K_{d,d}$. This provides asymptotic evidence for a conjecture of S. Friedland {\it et al.}. We also obtain an analogous result for independent sets of a fixed size…
For a fixed integer $k\geqslant 2$, let $G\in \mathcal{G}(n,p)$ be a simple connected graph on $n\rightarrow\infty$ vertices with the expected degree $d=np$ satisfying $d\geqslant c$ and $d^{k-1}= o(n)$ for some large enough constant $c$. We show that the asymptotical size of any maximal collection of edges $M$ in $G$ such that no two edges in $M$ are within distance $k$, which is called a distance $k$-matching, is between $ \frac{(k-1)n\log d}{4d^{k-1}}$ and $ \frac{k n \log d}{2d^{k-1}}$. We also design a randomized greedy algorithm to generate one large distance $k$-matching in $G$ with asy…
A matching $M$ of a graph $G$ is maximal if it is not a proper subset of any other matching in $G$. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In particular, almost nothing is known about their enumerative properties. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting linear polymers: polyspiro chains and benzenoid chains. We also analyze the asymptotic behavior of those sequences and determine the extremal cases.
Our previous paper applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs to the space of random injections from an $m$-element set to an $n$-element set. Equivalently, the same story can be told about the space of random matchings in $K_{n,m}$. Now we show how the cited version of the Lovász Local Lemma applies to the space of random matchings in $K_{2n}$. We also prove tight upper bounds that asymptotically match the lower bound given by the Lovász Local Lemma. As a consequence, we give new proofs to results on the enumeration of $d$-regular graphs. The t…
We prove that the number of perfect matchings in ${\mathcal G}(n,d)$ is asymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and $d=O(n^{1/7}/\log^2 n)$. This is the first distributional result of spanning subgraphs of ${\mathcal G}(n,d)$ when $d\to\infty$. Moreover, we prove that ${\mathcal G}(n,d-1)$ and ${\mathcal G}(n,d)$ can be coupled so that ${\mathcal G}(n,d-1)$ is a subgraph of ${\mathcal G}(n,d)$ with high probability when $d\to\infty$ and $d=o(n^{1/3})$. Further, if $d=Ω(\log^7 n)$, $d=O(n^{1/7}/\log^2n)$, and $d\le d'\le n-1$ then ${\mathcal G}(n,d)$ and ${\mathcal…
The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph, which is proposed first by Gutman and Wagner [The matching energy of a graph, Discrete Appl. Math. 160 (2012) 2177--2187]. And they gave some properties and asymptotic results of the matching energy. In this paper, we characterize the trees with $n$ vertices whose complements have the maximal, second-maximal and minimal matching energy. Further, we determine the trees with a perfect matching whose complements have the second-maximal matching energy. In particular, show that the …
For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures…
In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erdös-Rényi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.
We study the asymptotic distribution of the number of matchings of size $\ell=\ell(n)$ in $G(n,p)$ for a wide range of $p=p(n)\in (0,1)$ and for every $1\le \ell\le \lfloor n/2\rfloor$. We prove that this distribution changes from normal to log-normal as $\ell$ increases, and we determine the critical value of $\ell$, as a function of $n$ and $p$, at which the transition of the limiting distribution occurs.
We study minimal vertex covers and maximal matchings on trees. We pay special attention to the corresponding backbones i.e. these vertices that are occupied and those that are empty in every minimal vertex cover (resp. these egdes that are occupied and those that are empty in every maximal matching). The key result in our approach is that for trees, the backbones can be recovered from a particular tri-coloring which has a simple characterization. We give applications to the computation of some averages related to the enumeration of minimal vertex covers and maximal matchings in the random labe…
