We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform hypergraphs on $n$ vertices provided that $dn\le c\binom{n}{k}$ for a constant $c>0$, and $3 \leq k < n^C$ for any $C<1/9.$ Our results relate the degree sequence of a random $k$-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.
A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main resu…
For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ \frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \frac32})$. We also find the probability that a random $r$-uniform lin…
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edg…
We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and we strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.
An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i, j\},\ i\neq j$. For every $r\geq 3$ and $\ell\geq 4$, we show that there exists a constant $C$ depending on $r$ and $\ell$ such that the number of linear $r$-graphs of girth $\ell$ is at most $2^{Cn^{1+1/\lfloor \ell/2\rfloor}}$. Furthermore, we extend the result for $\ell=4$, proving that there exists a constant $C$ depending on …
In this short note, we compute the precise asymptotics for the number of contingency tables with non-uniform margins. More precisely, for parameter $n,δ, B,C>0$, we consider the set of matrices whose first $[n^δ]$ rows and columns have sum $[BCn]$ and the rest $n$ rows and columns have sum $[Cn]$. We compute the precise asymptotics of the cardinality of this set when $B<B_c=1+\sqrt{1+1/C}$ using the maximal entropy methods developed by Barvinok and Hartigan. The only contribution of this note is a detailed expansion of the determinant of quadratic forms in asymptotic formulas.
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.
In this paper we prove the necessity of the main sufficient condition of Meinardus for sub exponential rate of growth of the number of structures, having multiplicative generating functions of a general form and establish a new necessary and suffcient condition for normal local limit theorem for aforementioned structures. The latter result allows to encompass in our study structures with weights having gaps in their supports.
We show a first-order asymptotics result for the number of galled networks with $n$ leaves. This is the first class of phylogenetic networks of {\it large} size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of a galled networks with $n$ leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) which was devised for the purpose of (exactly) counting galled networks. Moreover, an old …
In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\inf…
Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic behaviour of such integrals, we establish explicit bounds on the exponentials of complex martingales. Those bounds applied to the case of truncated normal distributions are precise enough to include and extend many enumerative results of Barvinok, Canfield, Gao, Greenhill, Hartigan, Isaev, McKay, Wang, Wormald, and others. Our method applies to sums as well as inte…
The paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of the paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers…
A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are $2^{(1+o(1))\text{ex}(n,H)}$ such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs $H$. We make a first attempt at addressing this enumeration problem for a general bipartite graph $H$. We show that an upper bound of $2^{O…
A fundamental problem in pattern avoidance is describing the asymptotic behavior of the extremal function and its generalizations. We prove an equivalence between the asymptotics of the graph extremal function for a class of bipartite graphs and the asymptotics of the matrix extremal function. We use the equivalence to prove several new bounds on the extremal functions of graphs. We develop a new method to bound the extremal function of hypergraphs in terms of the extremal function of their associated multidimensional matrices, improving the bound of the extremal function of $d$-permutation hy…
Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given descent set). We then use this formula to show that, for almost all sets $I \subseteq [n-1]$, the fraction of size-$n$ permutations with descent set $I$ which are $n$-cycles is asymptotically $1/n$. As a special case, we recover a result of Stanley for alternating cycles. We also use our formula to count the cycles that do not have two consecutive descents.
Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have been previously enumerated by length and perimeter (Bousquet-Mélou, Schwerdtfeger; 2010). We consider the enumeration of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hyper…
Sharp asymptotic bounds for uniform union-free hypergraphs
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究中心）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（韩国基础科学研究院极值组合与概率组）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性期望前沿科学中心）
This work is a follow-up of the article [Proc.\ London Math.\ Soc.\ 119(2):358--378, 2019], where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$ vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} ρ^{-n} n!$, where $g>0$ is a constant and $ρ\approx 0.24377$ is the radius of convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of several classes of…
Equations in finite semigroups: Explicit enumeration and asymptotics of solution numbers
研究了一般单变量半群方程$X^\al=X^\be$的解的个数，以及方程组$X^2=X, Y^2=Y, XY=YX$在$H\wr T_n$中的解的个数，其中$H\wr T_n$是任意有限群$H$与全变换半群$T_n$在$n$上的圈积.对于这些解决方案的数量，我们提供明确的精确公式，以及渐近估计。我们关于第一个问题的结果推广了Harris和Schoenfeld（J. Combin. Theory Ser. A 3（1967），122-135）关于$T_n$中幂等元的个数，以及Dress和第二作者的部分结果（Adv. in Math. 129（1997），188-221）。其中采用的渐近工具是海曼的方法估计系数的解析函数和泊松求和公式。
Let $\mathcal{F}$ be a family of $3$-uniform linear hypergraphs. The linear Turán number of $\mathcal F$ is the maximum possible number of edges in a $3$-uniform linear hypergraph on $n$ vertices which contains no member of $\mathcal{F}$ as a subhypergraph. In this paper we show that the linear Turán number of the five cycle $C_5$ (in the Berge sense) is $\frac{1}{3 \sqrt3}n^{3/2}$ asymptotically. We also show that the linear Turán number of the four cycle $C_4$ and $\{C_3, C_4\}$ are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte. We establish a connecti…
For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsym…
