Lattice Path Enumeration

Enumeration of $k$-Fibonacci Paths using Infinite Weighted Automata

In this paper, we introduce a new family of generalized colored Motzkin paths, where horizontal steps are colored by means of $F_{k,l}$ colors, where $F_{k,l}$ is the $l$th $k$-Fibonacci number. We study the enumeration of this family according to the length. For this, we use infinite weighted automata.

Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees

We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Faà di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results …

Redundant generating functions in lattice path enumeration

A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a surprising occurrence of Catalan numbers in counting paths that stay below the line y = 2x. In the second application we prove a conjecture of Niederhausen and Sullivan.

enumeration of weighted paths on a digraph and block hook determinant

In this article, we evaluate determinants of block hook matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.

Enumeration of paths and cycles and e-coefficients of incomparability graphs

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations.

Enumeration of Deutsch paths by adding the adding-a-new-slice method and applications

A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas due to Emeric Deutsch. We use the adding-a-new-slice technique and the kernel method to compute the number of maximal runs of up-step runs of length 1 and a subclass of Deutsch paths satisfying a condition that was stipulated by R. Stanley for Dyck paths.

Enumeration of Some Closed Knight Paths

The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence classes of closed knight paths of length 4, exactly 25 equivalence classes of closed knight paths of length 6 and exactly 478 equivalence classes of closed knight paths of length 8.

讲座厅序多胞形中的加权格点枚举

Weighted lattice point enumeration in lecture hall order polytopes

本文研究讲座厅序多胞形中的加权格点枚举，发现其半开多胞形对多项式权重封闭，并证明有根树的加权$h^\ast$-多项式实根，补充了先前结果。

Lattice Path Enumeration and Its Applications in Representation Theory

In this paper, we enumerate lattice paths with certain constraints and apply the corresponding results to develop formulas for calculating the dimensions of submodules of a class of modules for planar upper triangular rook monoids. In particular, we show that the famous Catalan numbers appear as the dimensions of some special modules; we also obtain some combinatorial identities

Chain enumeration, partition lattices and polynomials with only real roots

The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen--Macaulay posets. As a result, new families of convex polytopes whose face lattices have real-rooted chain polynomials are presented. An application to the face enumeration of the second barycentri…

Enumeration of lattice 3-polytopes by their number of lattice points

We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216,453 of them). In order to achieve this we prove that if $P$ is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enume…

Enumeration of Fuss-Schröder paths

In this paper we enumerate the number of $(k, r)$-Fuss-Schröder paths of type $λ$. Y. Park and S. Kim studied small Schröder paths with type $λ$. Generalizing the results to small $(k, r)$-Fuss-Schröder paths with type $λ$, we give a combinatorial interpretation for the number of small $(k, r)$-Fuss-Schröder paths of type $λ$ by using Chung-Feller style. We also give two sets of sparse noncrossing partitions of $[2(k + 1)n + 1]$ and $[2(k + 1)n + 2]$ which are in bijection with the set of all small and large, respectively, $(k, r)$-Fuss-Schröder paths of type $λ$.

六边形圆堆积中的路径计数

Enumeration of paths in a hexagonal circle packing

针对六边形圆堆积中的路径，本文用函数方程、核方法等推导其生成函数、渐近结果，建立与三类经典路径的双射，证明相关阵列为Riordan阵列并得到连分数展开。

Weighted Lattice Paths Enumeration by Gaussian Polynomials

The Gaussian polynomial in variable $q$ is defined as the $q$-analog of the binomial coefficient. In addition to remarkable implications of these polynomials to abstract algebra, matrix theory and quantum computing, there is also a combinatorial interpretation through weighted lattice paths. This interpretation is equivalent to weighted board tilings, which can be used to establish Gaussian polynomial identities. In particular, we prove duals of such identities and evaluate related sums.

On rational Dyck paths and the enumeration of factor-free Dyck words

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by factor-free words. This bijection leads to a new statistic based on the reducibility level of the paths for which we provide a corresponding formula. On the other hand, we prove an inverse relation for certain sequences defined via partial Bell polynomials, and we use it to derive a formula for the enumeration of factor-free words. In addition, we give alterna…

On Enumeration of Dyck--Schröder Paths

We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along the diagonal itself. Our analysis leads to explicit generating functions and an efficient way to compute terms of many sequences in the Online Encyclopedia of Integer Sequences, proposed by Clark Kimberling almost two decades ago.

On Enumeration of Dyck Paths with colored hills

We continue to investigate the properties of the earlier defined functions fm and gm, which depend on an initial arithmetic function f0. In this papers values of f0 are the Fine numbers. We investigate functions fi; gi; (i = 1; 2; 3; 4). For each function, we derive an explicit formula and give a combinatorial interpretation. It appears that g2 and g3 are well-known combinatoric object called the Catalan triangles. We finish with an identity consisting of ten items.

Lattice path enumeration for semi-magic squares by Latin rectangles

Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.

通过双色 Motzkin 路径精细枚举两行集值标准表

Refined enumeration of two-rowed set-valued standard tableaux via two-coloured Motzkin paths

本文通过建立两行集值标准表与双色 Motzkin 路径的双射，结合生成函数与拉格朗日反演公式，给出了按各行条目数精细计数的显式公式。

Enumeration of partial Lukasiewicz paths

Łukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length prefixes (resp. suffixes) of these paths ending at height $k\geq 0$ with a given type of step. We make a similar study for prefixes of height at most $t\geq 0$. Using the explicit forms for the paths of bounded height, we evaluate the average height asymptotically. For fixed $k$ and $n\to\infty$, this quantity behaves as $\sqrt{πn}$. Finally we study (in the same way)…

整数组合关联的多面体格点计数

Lattice point enumeration of polytopes associated to integer compositions

本文证明了Chapoton关于整数组合关联多面体h向量的猜想，并给出了gamma向量非负性及h*多项式组合解释。

Enumeration of colored Dyck paths via partial Bell polynomials

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in an unified manner.

Fibonacci 多联骨牌：精细枚举与 Dyck 路径双射

Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections

本文研究 Fibonacci 多联骨牌的按面积和半周长枚举，通过生成函数得到含 Catalan 数的精细公式，并建立与带标签 Dyck 路径的双射，将统计量转化为峰与返回，还研究等高连续列并得到 Motzkin 数控制的精细化。

Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths

Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3]. In the paper we enumerate the terms of the OEIS A036991, Dyck numbers, and construct a concomitant bijection with symmetric Dyck paths. In the case of binary coding of Dyck paths we work with compact natural numbers after removing leading zeros. Analysis of binary suffixes, allowed us to obtain a bijection between arbitrary A036991 terms and symmetric A036991 terms which encode symmetric Dyck paths. The bijection generates a forest of unary non-interse…

Asymptotic lattice path enumeration using diagonals

This work presents new asymptotic formulas for family of walks in Weyl chambers. The models studied here are defined by step sets which exhibit many symmetries and are restricted to the first orthant. The resulting formulas are very straightforward: the exponential growth of each model is given by the number of steps, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. These expressions are derived by analyzing the singular variety of a multivariate rational function whose diagonal counts the lattic…

Schröder Paths and Pattern Avoiding Partitions

In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schröder paths of semilength $n$ without peaks at even level. As a consequence, the refined enumeration of 12312-avoiding (resp. 12321-avoiding) partitions according to the number of blocks can be reduced to the enumeration of certain Schröder paths according to the number of peaks. Furthermore, we get the enumeration of irreducible 12312-avoiding (resp. 12321-avoiding) partitions, which are closely related to skew Dyck paths.

Enumerations of humps and peaks in $(k,a)$-paths and $(n,m)$-Dyck paths via bijective proofs

Recently Mansour and Shattuck studied $(k,a)$-paths and gave formulas that relate the total number of humps (peaks) in all $(k,a)$-paths to the number of super $(k,a)$-paths. These results generalize earlier results of Regev on Dyck paths and Motzkin paths. Their proofs are based on generating functions and they asked for bijective proofs for their results. In this paper we first give bijective proofs of Mansour and Shattuck's results, then we extend our study to $(n,m)$-Dyck paths. We give a bijection that relates the total number of peaks in all $(n,m)$-Dyck paths to certain free $(n,m)$-pat…

Refined Lattice Path Enumeration and Combinatorial Reciprocity

It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fuß-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps. We give exact formulas for the number of such paths with fixed height, fixed number of returns and (i) fixed number of valleys, (ii) fixed number of valleys with $x$-coordinate divisible by $m$ and (iii) fixed number of valleys with $x$-coordinate not divisible by $m$. The enumeration (ii) combinatorially realizes the $H$-triangle appearing in a recent article of…

通过Motzkin路径和四分仪游走对区间闭集进行枚举

Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks

本文通过构造双射分别将两条链之积偏序集的区间闭集映射到双色Motzkin路径，将截断矩形偏序集的区间闭集映射到四分仪游走，从而求得了对应的生成函数与函数方程。

Enumeration of symmetry classes of convex polyominoes in the square lattice

This paper concerns the enumeration of rotation-type and congruence-type convex polyominoes on the square lattice. These can be defined as orbits of the groups C4, of rotations, and D4, of symmetries of the square acting on (translation- type) polyominoes. In virtue of Burnside's Lemma, it is sufficient to enumerate the various symmetry classes (fixed points) of polyominoes defined by the elements of C4 and D4. Using the Temperley--Bousquet-Melou methodology, we solve this problem and provide explicit or recursive formulas for their generating functions according to width, height and area. We …

Enumeration of closed random walks in the square lattice according to their areas

We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple argument. It is concluded that the number of walks of length $n$ and area $s$ equals to the coefficient of $z^s$ in the expression $(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a special group ring $R[x,y,z]$. A polynomial time algorithm for calculating these values, is then concluded. Finally, the provided algorithm and the results of impleme…

Signed Enumeration of Upper-Right Corners in Path Shuffles

We resolve a conjecture of Albert and Bousquet-Melou enumerating quarter-plane walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed upper-right-corner count statistic. We find its distribution over planar walks with any choice of fixed horizontal and vertical projections. Additionally, we prove that the polynomial counting loops with a fixed horizontal and vertical projection according to the absolute value of their signed upper-right-corner count is $(x+1)$-positive. Finally, we conjecture an equivalenc…

Enumeration in convex geometries and associated polytopal subdivisions of spheres

We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of faces in these spheres can be obtained from the defining lattices in a manner analogous to the relation between arrangements of hyperplanes and their underlying geometric intersection lattices.

Lattice Walk Enumeration

Trying to enumerate all of the walks in a 2D lattice is a fun combinatorial problem and there are numerous applications, from polymers to sports. Computers provide a wonderful tool for analyzing these walks; we provide a Maple package for automatically describing generating functions of walks restricted to any step set in a 2D lattice. We always obtain a closed system of relations for generating functions of walks that are bounded, semi-bounded, or unbounded. For bounded walks, this leads to explicit rational solutions! For semi-bounded or unbounded walks, we may get lucky and obtain algebraic…

Enumeration of Hybrid Domino-Lozenge Tilings

We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every third diagonal drawn in. We also obtain a generalization of Douglas' Theorem on the number of tilings of a family of regions of the square lattice with every second diagonal drawn in.

A continuous analogue of lattice path enumeration

Following the work of Cano and Diaz, we consider a continuous analog of lattice path enumeration. This allows us to define a continuous version of any discrete object that counts certain types of lattice paths. We define continuous versions of binomials and multinomials, and describe some identities and partial differential equations they satisfy. Finally, we illustrate a general process to recover discrete combinatorial quantities from their continuous analogs.

Overview of some general results in combinatorial enumeration

This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicit…

幺模矩阵与通过帕斯卡三角形的格路枚举

Unimodular matrices and lattice paths enumeration via Pascal's triangle

研究一类由二项式系数定义的矩阵，证明其行列式恒为1，揭示了帕斯卡三角形与格路计数之间的深刻联系。

Enumeration of edges in some lattices of paths

We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder and Grand Schröder lattices. Finally, we give a general formula for the number of edges in an arbitrary Young lattice (which can be interpreted in a natural way as a lattice of paths).

On the enumeration of Hoppy's walks

The enumeration of k-Dyck paths ending at level j after m up-steps, where the last step is an up-step, is given as a sum, improving on a previous formula given by Deng and Mansour.

Exact enumeration of self-avoiding walks on BCC and FCC lattices

Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents $γ$ and $ν$ which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice $μ=6.530520(20)$, $A=1.1785(40)$, and $D=1.0864(50)$, and for the FCC lattice $μ=10.037075(20)$, $A=1.1736…

The $\mathbf{uvu}$-avoiding $(a,b,c)$-Generalized Motzkin paths with vertical steps: bijections and statistic enumerations

A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. In this paper, we first give bijections between the set of $\mathbf{uvu}$-avoiding …

Toric arc schemes and $q$-enumeration of lattice points

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.

Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics

The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded pla…

闭合骑士路径的构造、分类与枚举算法

Algorithms for Construction, Classification and Enumeration of Closed Knight's Paths

本文提出了构造长度不超过16的闭合骑士路径的两种算法及等价分类方法，并据此枚举了无限制和非相交骑士路径，展示了所得结果。

On embedded trees and lattice paths

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of embedded trees and map enumeration. The aim of this note is to apply this method to three problems. First, we discuss a general family of embedded binary trees, trying to unify and summarize several enumeration results for binary tree families, and also to add new results. Second, we discuss the family of embedded $d$-ary trees, embedded in the plane in a natural way. Third, we show that several enumeration problems concerning simple families of lattice pa…

零化度为$k$且包含$r$个可比较可约元的格的计数

Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements

本文针对Birkhoff计数$n$元有限格的公开问题，先计数$n≥4$元、零化度$k≥1$且含$2≤r≤2k$个可比较可约元的非同构RC-格，再计数$n≥4$元、零化度$k≥1$的非同构RC-格，完成$n$元非同构RC-格的枚举。

Enumeration of Various Animals on the Triangular Lattice

In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex polyiamonds with respect to their perimeter. In particular, we show that the number of (A) baryiamonds of perimeter $n$ is asymptotically $$\frac{(ξ+1)^2\sqrt{ξ^4+ξ^3-2ξ+1}}{2\sqrt{πn^3}}ξ^{-n-2},…

Enumeration of walks on lattices. I

This work develops a methodical approach to counting of walks on cartesian products, biproducts, symmetric and exterior powers and bipowers, Schur operations, coverings and semicoverings of weighted graphs. For weight and root lattices of semisimple Lie algebras, this approach allows us to compute various combinatorial and representation-theoretical constants, in particular, the number of plane symplectic wave graphs with given number of vertices.

Enumeration of saturated chains in Dyck lattices

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.

有限圆柱格图中连通集枚举的研究

On the enumeration of connected sets in finite cylindrical lattice graphs

本文研究有限圆柱格图中连通集的枚举问题，利用多步递推公式得到 $m\le 7$ 时 $N(C_m\times P_n)$ 的枚举公式，并通过转移矩阵方法给出一般笛卡尔积图连通集数的紧下界及若干格图的渐近结果。

某些树形多面体的格点计数

Lattice point enumeration of some arbor polytopes

本文研究了某些树形多面体的格点计数问题，通过新的组合模型证明了其Ehrhart多项式为魔术正，并推导了h*多项式的实根性。

Enumeration of lattice polytopes by their volume

A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional s…

A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice

We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.