Vladimir Potapov, Denis Krotov
The asymptotic form of the number of n-quasigroups of order 4 is $3^{n+1} 2^{2^n +1} (1+o(1))$. Keywords: n-quasigroups, MDS codes, decomposability, reducibility.
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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…
Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a recurrent formula for Q(n,4). We prove that for all $n\geq 2$ and $k\geq 5$ the following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k) \leq {c_k(k-2)^{n}} $, where $c_k$ does not depend on $n$. So, the upper asymptotic bound for $Q(n,k)$ is improved for any $k\geq 5$ and the lower bound is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop, asymptotic estimate, component, latin trade.
On the reconstruction of $n$-quasigroups of order $4$ and upper bounds on their number
本文研究4阶n元拟群的重构问题，确定其数量的渐近式并给出上界，是相关代数结构计数研究的成果。
A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$…
(1) We prove that, provided n>=4, a permutably reducible n-ary quasigroup is uniquely specified by its values on the n-ples containing zero. (2) We observe that for each n,k>=2 and r<=[k/2] there exists a reducible n-ary quasigroup of order k with an n-ary subquasigroup of order r. As corollaries, we have the following: (3) For each k>=4 and n>=3 we can construct a permutably irreducible n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>3 has double-exponential growth as n tends to infinity; it is greater than exp exp(n ln[k/3]) if k>=6, and exp exp(n (ln 3)/3 - 0.44)…
We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes, and extend previously published results to include information on the number and classes of medial totally symmetric quasigroups for all orders up to and including 16.
Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commuta…
An $n$-ary quasigroup $f$ of order $q$ is an $n$-ary operation over a set of cardinality $q$ such that the Cayley table of the operation is an $n$-dimensional latin hypercube of order $q$. A transversal in a quasigroup $f$ (or in the corresponding latin hypercube) is a collection of $q$ $(n+1)$-tuples from the Cayley table of $f$, each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011. An $n$-ary quasigroup $f$ is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is …
Associativity conditions for linear quasigroups and equivalence relations on binary trees
本文通过基于叶子深度模整数的二叉树等价关系，刻画线性拟群的括号恒等式，其等价类数构成卡特兰数变体，即拟群的结合谱。
On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups
该研究针对奇素数幂q下由二次运算定义的拟群Q_{a,b}，推导了使其为极大非结合拟群的参数对(a,b)的数量σ(q)的渐近极限，给出q模4不同余时的两个常数α、β。
An algebraic system from a finite set $Σ$ of cardinality $k$ and an $n$-ary operation $f$ invertible in each argument is called an $n$-ary quasigroup of order $k$. An autotopy of an $n$-ary quasigroup $(Σ,f)$ is a collection $(θ_0,θ_1,...,θ_n)$ of $n+1$ permutations of $Σ$ such that $f(θ_1(x_1),...,θ_n(x_n))\equiv θ_0(f(x_1,\ldots,x_n))$. We show that every $n$-ary quasigroup of order $4$ has at least $2^{[n/2]+2}$ and not more than $6\cdot 4^n$ autotopies. We characterize the $n$-ary quasigroups of order $4$ with $2^{(n+3)/2}$, $2\cdot 4^n$, and $6\cdot 4^n$ autotopies.
Autoparatopisms of Quasigroups and Latin Squares
研究quasigroups和拉丁方的自paratopism性质，确定n≤17时Par(n)的结构，并分析n趋于无穷时Par(n)在所有paratopism中的比例。
We say that $(x,y,z)\in Q^3$ is an associative triple in a quasigroup $Q(*)$ if $(x*y)*z=x*(y*z)$. Let $a(Q)$ denote the number of associative triples in $Q$. It is easy to show that $a(Q)\ge |Q|$, and we call the quasigroup maximally nonassociative if $a(Q)= |Q|$. It was conjectured that maximally nonassociative quasigroups do not exist when $|Q|>1$. Drápal and Lisoněk recently refuted this conjecture by proving the existence of maximally nonassociative quasigroups for a certain infinite set of orders $|Q|$. In this paper we prove the existence of maximally nonassociative quasigroups for a mu…
Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) 2036029552582883134196099 main classes of Latin squares of order 11; (ii) 6108088657705958932053657 isomorphism classes of one-factorizations of $K_{11,11}$; (iii) 12216177315369229261482540 isotopy classes of Latin squares of order 11; (iv) 1478157455158044452849321016 isomorphism classes of loops of order 11; and (v) 19464657391668924966791023043937578299025 isomorphism classes of quasigroups of order 11. The enumeration is constructive for the 1151666…
We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-$n$ Latin squares with no intercalate (i.e., no $2\times2$ Latin subsquare) is at least $(e^{-9/4}n-o(n))^{n^{2}}$. Equivalently, $\mathbb{P}\left[\mathbf{N}=0\right]\ge e^{-n^{2}/4-o(n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}$, where $\mathbf{N}$ is the number of intercalates …
We calculate the asymptotic value of the choice number of complete multi-partite graphs.
In this paper, the asymptotic formulas for Eulerian numbers, refined Eulerian numbers and the coefficients of descent polynomials are obtained directly from the spline interpretations of these numbers. Having related these numbers directly to B-splines [15], we can take advantage of many powerful spline techniques to derive various results of these numbers. The asymptotic formulas for the Eulerian numbers Ad;k agree with the previously known results which were given by L. Carlitz et al.(1972)[2] and S.Tanny (1973) [18], but the convergence order is much better. We also give the asymptotic repr…
Asymptotics for face numbers of certain Hanner polytopes, with applications
研究特定Hanner多面体的面数渐进行为，并接近满足FLM不等式的一类参数。
We determine the asymptotic behaviour of the number of the Eulerian circuits in undirected simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). We also prove some new properties of the Laplacian matrix.
We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.
We derive a functional relation between the generating functions of connected chord diagrams and 2-connected chord diagrams. This relation enables us to calculate an asymptotic expansion for the number of 2-connected chord diagrams on $n$ chords. The asymptotic information obtained from this expansion refines the last established results and provides a simple alternative for calculating the asymptotic behaviour of certain Green functions in Quenched QED and Yukawa theory in the context of quantum field theory.
In 1938 E. T. Bell introduced "The Iterated Exponential Integers". He proved that these numbers may be expressed by polynomials with rational coefficients. However, Bell gave no formulas for any of the coefficients except the trivial one, which is always 1. Our task has been to find the coefficient of the leading term, giving asymptotic information about these numbers.
We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In addition, we establish some new properties of the Laplacian matrix, as well as an estimate of a conditionality of matrices with the asymptotic diagonal predominance
We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We also compare our bounds with the existing bounds on the numbers of linear extensions of the corresponding posets.
Elementary asymptotics for the Stirling numbers of the second kind: The central range
通过初等方法推导第二类斯特林数的局部和中心极限定理，并得到Bell数及分布矩的有效渐近估计，同时基于有限差分方法给出沿多个方向的渐近展开。
We show that the number of prefix normal binary words of length $n$ is $2^{n-Θ((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.
Formulas and asymptotics of hypergraph Catalan numbers
本文研究了k-游走的渐进行为，通过另一种方法验证了Gunnells关于超图卡塔兰数的渐近公式，发现其增长由星形树上的k-游走数量决定。
P.L. Erdos and L.A. Szekely [Adv. Appl. Math. 10(1989), 488-496] gave a bijection between rooted semilabeled trees and set partitions. L.H. Harper's results [Ann. Math. Stat. 38(1967), 410-414] on the asymptotic normality of the Stirling numbers of the second kind translates into asymptotic normality of rooted semilabeled trees with given number of vertices, when the number of internal vertices varies. The Erdos-Szekely bijection specializes to a bijection between phylogenetic trees and set partitions with classes of size \geq 2. We consider modified Stirling numbers of the second kind that en…
It is known that the $S(n,k)$ Stirling numbers as well as the ordered Stirling numbers $k!S(n,k)$ form log-concave sequences. Although in the first case there are many estimations about the mode, for the ordered Stirling numbers such estimations are not known. In this short note we study this problem and some of its generalizations.
We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.
Asymptotic number of edge-colored regular graphs
本文推导了边着色正则图的渐近数量公式，并应用该结果计算了大随机k-正则图的c-边着色期望数量。
We study lattice walks in a Weyl chamber of type A with fixed or free end points. For lattice walk models with zero drift that may be counted by means of a reflection argument, we determine asymptotics for the number of such walks as their length tends to infinity. These models are equivalent to the lock step model and the random turns model of vicious walkers. As special cases, our main results include various asymptotic formulas found in the literature.
The generating function for the sequence A166861 in the OEIS (Euler transform of Fibonacci numbers) is Product_{k>0} 1/(1-x^k)^F(k), where F(k) are the Fibonacci numbers. This paper analyzes the more general generating function U(x) = Product_{k>0} 1/(1-x^k)^F(k+z), where z is a nonnegative integer, which provides asymptotics for the sequences A166861 (z=0), A200544 (z=1) and A260787 (z=2) in the OEIS.
Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is valid for any value of the number of parts. We obtain general asymptotic formulas for distinct partitions that are valid in a wider range of parameters than the existing asymptotic formulas, and we recover the known asymptotic results as special cases of our general formulas.
A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order $n$ there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow\infty$.
We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{λ/μ}$, where $f^{λ/μ}$ is the number of standard Young tableaux of skew shape $λ/μ$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.
We consider lattice walks in $\R^k$ confined to the region $0<x_1<x_2...<x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main results are asymptotic formulas for the total number of walks of length $n$ with either a fixed or a free end point for a general class of walks as $n$ tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2,...,n\}$ as $n$ tends to infinity, and asymp…
Lieanders are special cases of meanders and first appeared in connection with Lie algebras. Using the results from the author with E. Goujard, P. Zograf and A. Zorich, we prove a polynomial asymptotics for the number of lieanders with fixed composition sizes as the number of arches tend to infinity. The coefficients of the asymptotics are rational numbers divided by an even power of pi.
An asymptotically tight upper bound for the domination number of the $2$-token graph of path graphs
本文研究了路径图2-令牌图的支配数，得出其渐近紧上界为n²/10+Θ(n)。
S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic $m_1 \times \cdots \times m_d$ tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case $m_1 = \cdots = m_d$. We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effe…
We provide a sharp estimate for the asymptotic number of lattice zonotopes, inscribed in $[0,n ]^d$ when $n$ tends to infinity. Our estimate refines the logarithmic equivalent established by Barany, Bureaux, and Lund when the sum of the generators of the zonotope is prescribed. As we shall see, the exponential part of our estimate is composed of a polynomial of degree $d$ in $n^{1/(d+1)}$, and involves Riemann's zeta function and its non-trivial zeros. %Our analysis is based on a mapping between sums of coprime numbers and Eulerian polynomials. We also analyze some combinatorial properties of …
For the Legendre-Stirling numbers of the second kind asymptotic formulae are derived in terms of a local central limit theorem. Thereby, supplements of the recently published asymptotic analysis of the Chebyshev-Stirling numbers are established. Moreover, we provide results on the asymptotic normality and unimodality for modified Legendre-Stirling numbers.
For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. Thereby a supplement of the asymptotic analysis for these numbers is established.
The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.
Asymptotic Analysis of Central Binomiacci Numbers
该研究利用生成函数的奇点分析方法，对包含斐波那契数且推广了二项式系数的数列开展了渐近分析，丰富了相关组合数列的渐近性质研究。
We obtain formulas for the growth rate of the numbers of certain paths in infinite graphs built on the two-dimensional Eulerian graph. Corollaries are identities relating Stirling numbers of the first and second kinds.
This paper has been withdrawn by the author; a revised version is part of the author's phd-thesis "Quasi-logarithmic structures" (Zurich, 2007).
Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler's generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers, that count the number of subspaces in vector sp…
We derive asymptotic bounds for the ordinary generating functions of several classical arithmetic functions, including the Moebius, Liouville, and von Mangoldt functions. The estimates result from the Korobov-Vinogradov zero-free region for the Riemann zeta-function, and are sharper than those obtained by Abelian theorems from bounds for the summatory functions.
On large genus asymptotics of certain Hurwitz numbers
本文基于中心字符在交换操作上的值，研究了某些Hurwitz数的结构和大亏格渐进行为。
Asymptotics for the number of bipartite graphs with fixed surplus
本文利用二部随机图的近期编码，为具有固定剩余的二部图数量提供了概率公式，并推导出各类顶点数趋于无穷大时的渐近性。
The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are studied. It is shown, in particular, that $log I_d\asypt d^2$. Other related problems and their higher dimensional generalisations are discussed.
We identify a relationship between a random walk on a certain Euclidean lattice and incidence matrices of balanced incomplete block designs. We then compute the return probability of the random walk and use it to obtain the asymptotic number of BIBD incidence matrices (as the number of columns increases). Our strategy is similar in spirit to the one used by de Launey and Levin to count partial Hadamard matrices.
Inequalities and asymptotics for hook numbers in restricted partitions
本文研究了受限划分中钩数的渐进行为，通过渐近公式证明了奇数部分划分在平均意义上比不同部分划分有更多钩数。
I present an asymptotic formula for the Takeuchi numbers $T_n$. In particular, I give compelling numerical evidence and present a heuristic argument showing that $$T_n\sim C_T B_n\exp{1\over2}{W(n)}^2$$as $n$ tends to infinity, where $B_n$ are the Bell numbers, W(n) is Lambert's $W$ function, and $C_T=2.239...$ is a constant. Moreover, I show that the method presented here can be generalized to derive conjectures for related problems.
Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions determined by $A \times A \subset \mathbb{F}_p^2$. The key idea is to reduce the problem to counting the number of solutions to the bilinear Diophantine equation $ad+bc=p$ in variables $1\le a,b,c,d\le n$; our asymptotic formula for the number of solutions is of independent interest.
The uniform asymptotics for real double Hurwitz numbers with triple ramification II: lower bounds and asymptotics
本文研究了具有三重分支的实双Hurwitz数的均匀渐近行为，推导了其统一下界并证明了与复类似物的对数等价性。
A $k$-threshold function on a rectangular grid of size $m \times n$ is the conjunction of $k$ threshold functions on the same domain. In this paper, we focus on the case $k=2$ and show that the number of two-dimensional 2-threshold functions is~$\dfrac{25}{12π^4} m^4 n^4 + o(m^4n^4)$.
Asymptotics for partitions over the Fibonacci numbers and related sequences
本文结合经典分拆理论与 Fibonacci Dirichlet 级数的新结果，首次给出整数表示为非互异斐波那契数之和的方式数 $p_F(n)$ 的渐近公式，并将类似结果推广到一般线性递推序列。
We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on the distribution of branch points. We use it to prove, under certain conditions, existence of real Hurwitz covers as well as logarithmic equivalence of real and classical Hurwitz numbers. The lower bound is based on the \enquote{tropical} computation of real Hurwitz numbers in arXiv:1412.4235.
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