三维空间中带对角步格路的一个注记

Jr.Stocks讨论了从(0,0,0)到(n,n,n)的带对角步格路的计数问题.本文给出了[4]中主要结果的简单公式,并将其推广到了一般情形.     

Counting Lattice Paths with an Infinite Step Set and Special Access

We count lattice paths that remain in the first quadrant. A path can come from only finitely many lattice points, and if no further restrictions apply, can go to infinitely many others. By “further restrictions” we mean a boundary line above which the paths may have to stay. Access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set, which also may be infinite. Our approach to explicit solutions of such enumeration problems is via Sheffer polynomials and functionals, using results of the Umbral Calculus.     

Plethysm and Lattice Point Counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).     

两个带变时间步的两重网格算法的稳定性分析

对用于求解非线性发展方程的两个带变时间步的两重网格算法，对空间变量用有限元离散，对时间变量分别用一阶精度Euler显式和二阶精度半隐式差分格式离散，然后构造两重网格算法，通过深入的稳定性分析，得出本文的算法优于标准全离散有限元算法。     

A Note on Counting Homomorphisms of Paths

We obtain two identities and an explicit formula for the number of homomorphisms of a finite path into a finite path. For the number of endomorphisms of a finite path these give over-count and under-count identities yielding the closed-form formulae of Myers. We also derive finite Laurent series as generating functions which count homomorphisms of a finite path into any path, finite or infinite.     

Counting Lattice Points in The Sphere

We consider the error term which occurs in the counting of lattice points in a sphere ofradius R. By considering second and third power moments, weprove that . An upper bound for the gap between the sign changes of P₃(R) is also proved.1991 Mathematics Subject Classification 11P21.     

Counting Dyck Paths by Area and Rank

The set of Dyck paths of length 2n inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: area (the area under the path) and rank (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting (q, t)-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: One refines an identity of Carlitz and Riordan; the other refines the notion of γ-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the n-cycle (12· · ·n) in the absolute order on the symmetric group.     

Counting Boundary Paths for Oriented Percolation Clusters

We shall use analytic methods to prove bounds on the analogue of the connective constant for the oriented percolation on ℤ². ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 1–28, 1999     

Lattice Paths and Positive Trigonometric Sums

A trigonometric polynomial generalization to the positivity of an alternating sum of binomial coefficients is given. The proof uses lattice paths, and identifies the trigonometric sum as a polynomial with positive integer coefficients. Some special cases of the q -analogue conjectured by Bressoud are established, and new conjectures are given. January 22, 1997. Date revised: July 9, 1997.     

Counting d -Step Paths in Extremal Dantzig Figures

The d -step conjecture asserts that every d -polytope P with 2d facets has an edge-path of at most d steps between any two of its vertices. Klee and Walkup showed that to prove the d -step conjecture, it suffices to verify it for all Dantzig figures (P, w ₁ , w ₂ ) , which are simple d -polytopes with 2d facets together with distinguished vertices w ₁ and w ₂ which have no common facet, and to consider only paths between w ₁ and w ₂ . This paper counts the number of d -step paths between w ₁ and w ₂ for certain Dantzig figures (P, w ₁ , w ₂ ) which are extremal in the sense that P has the minimal and maximal vertices possible among such d -polytopes with 2d facets, which are d ² - d + 2 vertices (lower bound theorem) and vertices (upper bound theorem), respectively. These Dantzig figures have exactly 2 ^d-1 d -step paths. Received September 24, 1995, and in revised form December 12, 1995, and April 8, 1996.     

峰严格递增的Dyck路的计数

本文考虑了由最高峰的高度为m,并且峰的高度沿着Dyck路严格递增的所有Dyck路组成的集合,即集合Dm的子集的计数问题．利用双射、生成树以及Riordan阵的方法来对集合Dm的一些子集进行计数,得到了一些以经典的序列如Catalan数、Narayana数、Motzkin数、Fibonacci数、Schroder数以及第一类无符号Stirling数来计数的组合结构．特别地,我们给出了两个新的Catalan结构,它们并没有明显地出现在Stanley关于Catalan结构的列表中．     

Seven (Lattice) Paths to Log-Convexity

Three new methods for proving log-convexity of combinatorial sequences are presented. Their implementation is demonstrated and their performance is compared with four more familiar approaches in the context of sequences that enumerate various classes of lattice paths.     

Pairs of Lattice Paths and Positive Trigonometric Sums

Ismail et al. (Constr. Approx. 15:69–81, 1999) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.     

Cartesian Lattice Counting by the Vertical 2-sum

Order - A vertical 2-sum of a two-coatom lattice L and a two-atom lattice U is obtained by removing the top of L and the bottom of U, and identifying the coatoms of L with the atoms of U. This...     

Counting Lattice Points by Means of the Residue Theorem

We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.     

与格路有关的组合恒等式

通过研究格路径的性质得到一类组合恒等式的通式,代入不同的参数给出已有的一些组合恒等式新的简洁证明,并得到一些新的组合恒等式.最后推广得到多项式系数的恒等式.     

Counting Growth Types of Automorphisms of Free Groups

Given an automorphism of a free group Fn, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e, d, R) may be realized by an automorphism of Fn. In particular, the inequality e ￡ \frac3n-24{{e \leq \frac{3n-2}{4}}} (due to Levitt–Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism.