Enumeration of paths in Young--Fibonacci graph

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.

Ergodicity of the Martin boundary of the Young--Fibonacci graph. I

Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $β\in(0,1]$ (the case $β=0$ corresponds to the Plancherel measure). In this article we prove the statements which correspond to the case $β=1$.

数值半群枚举中的次 Fibonacci 行为

Sub-Fibonacci behavior in numerical semigroup enumeration

本文利用 Kunz 坐标和赋权图同态结果，证明了数值半群数量偏离完美指数增长因子的指数上下界，获得了已知最精确的渐近界并揭示了次 Fibonacci 行为。

Ergodicity of the Martin boundary of the Young--Fibonacci graph. II

Among central measures on the path space of the Young--Fibonacci lattice the so-called Plancherel measure has a special role. Its ergodicity was proved by Kerov and Gnedin. The goal of this cycle of two articles is to prove that remaining measures from the Martin boundary of this graph (which were described by Kerov and Goodman) are also ergodic. The measures are parametrized with an infinite word of digits 1 and 2 and the parameter $β\in(0,1]$ (the case $β=0$ corresponds to the Plancherel measure). In this article we finish the proof of their ergodicity using the statements proved in the firs…

Young--Fibonacci 图的跳跃图的中心测度

Martin boundary of the jump graph on subgraphs of the Young--Fibonacci graph

本文研究 Young--Fibonacci 图中至多含 k 个 2 的单词构成的子图所对应的跳跃图，并完整刻画了该图路径空间上的所有中心测度。

The Martin Boundary of the Young-Fibonacci Lattice

We find the Martin boundary for the Young-Fibonacci lattice YF. Along with the lattice of Young diagrams, this is the most interesting example of a differential poset. The Martin boundary construction provides an explicit Poisson-type integral representation of non-negative harmonic functions on YF. The latter are in a canonical correspondence with a set of traces on Okada locally semisimple algebra. The set is known to contain all the indecomposable traces. Presumably, all of the traces in the set are indecomposable, though we have no proof of this conjecture. Using a new explicit product for…

奇数索引斐波那契数 via 避免模式的排列

Odd-indexed Fibonacci numbers via pattern-avoiding permutations

本文研究了若干组合问题，其计数结果为奇数索引斐波那契数，包括特定类型的Dyck路径、块喷泉、有向列凸多边形和无交叉无嵌套的集合划分。通过构造到避免模式排列的双射，并推导生成函数以跟踪排列层面的统计量。

Enumeration of standard Young tableaux of certain truncated shapes

Unexpected product formulas for the number of standard Young tableaux of certain truncated shapes are found and proved. These include shifted staircase shapes minus a square in the NE corner, rectangular shapes minus a square in the NE corner, and some variations.

Young-Fibonacci 格与移位阶梯格上的 Ungar 博弈

Ungar Games on the Young-Fibonacci and the Shifted Staircase Lattices

本文研究 Young-Fibonacci 格与移位阶梯序理想格上的 Ungar 博弈，证明了 Defant、Kravitz 和 Williams 提出的两个猜想。

Enumeration of standard Young tableaux of shifted strips with constant width

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using of the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specially, $g_{n,4}$ is the $(2n-1)$-st Pell number.

Enumeration formulas for Young tableaux in a diagonal strip

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.

Young–Fibonacci图的r-微分版本上的中心测度

The Martin boundary of the $r$-differential version of the Young--Fibonacci graph

本文刻画了Young–Fibonacci图r-微分版本路径空间的Martin边界，并证明了相应中心测度的遍历性。

On the Enumeration of Skew Young Tableaux

We extend work of McKay, Morse, and Wilf by giving exact formulas and asymptotic formulas for the number of skew Young tableaux T in two situations: (1) the "inside shape" and total number of cells of T are fixed, and (2) the inside shape of T is fixed. The proofs use the theory of symmetric functions and estimates for irreducible characters of the symmetric group due to Thoma, Vershik, Kerov, and Biane.

树中的四种支配增长模式：强制、斐波那契计数、周期性和稳定性

Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability

本文研究了树中支配集的增长模式，揭示了不同构造方式对支配集数量的影响，并证明了完全二叉树中支配域的周期性规律及稳定性边界。

Enumeration and Structure of Trapezoidal Words

Trapezoidal words are words having at most $n+1$ distinct factors of length $n$ for every $n\ge 0$. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal wor…

Young--Fibonacci 格初等理论的不可判定性

Undecidability of the elementary theory of Young--Fibonacci lattice

本文针对与Young格性质相近的Young--Fibonacci格，参照A. Wires证明Young格初等理论不可判定的思路，通过在该格的初等理论中定义算术系统，证明其初等理论的不可判定性。

Residues modulo powers of two in the Young-Fibonacci lattice

We study the subgraph of the Young-Fibonacci graph induced by elements with odd $f$-statistic (the $f$-statistic of an element $w$ of a differential graded poset is the number of saturated chains from the minimal element of the poset to $w$). We show that this subgraph is a binary tree. Moreover, the odd residues of the $f$-statistics in a row of this tree equidistibute modulo any power two. This is equivalent to a purely number theoretic result about the equidistribution of residues modulo powers of two among the products of distinct odd numbers less than a fixed number.

Rook placements in Young diagrams and permutation enumeration

Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$. This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution are expressed using two operators $D$ and $E$ subject to the relation $DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel…

Asymptotic joint distribution of the extremities of a random Young diagram and enumeration of graphical partitions

An integer partition of $n$ is a decreasing sequence of positive integers that add up to $[n]$. Back in $1979$ Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In $1982$ Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability $0$. In $1997$ we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of con…

Enumeration of diagonally colored Young diagrams

In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews' work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.

A probabilistic approach for enumeration of certain Young tableaux

In this paper we establish an order statistics model of Young tableaux. Multiple integration over nested simplexes is applied to the enumeration of Young tableaux. A brief proof of Frobenius-Young's and Aitken's formulas is given. Partially standard Young tableaux and special truncated shapes including tableaux with a hole are discussed, the associated product formulas are given.

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.

On the enumeration of three-rowed standard Young tableaux of skew shape in terms of Motzkin numbers

The enumeration of standard Young tableaux (SYTs) of shape λ can be easily computed by the hook-length formula. In 1981, Amitai Regev proved that the number of SYTs having at most three rows with n entries equals the nth Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of skew shape λ/(2, 1) over all partitions λ having at most three parts with n entries is the difference of two Motzkin numbers, M_{n-1} - M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computer program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of skew shape …

Enumeration of Dyck paths with air pockets

We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities and asymptotic expectations of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for non-decreasing Dyck paths with air pockets.

On the enumeration of d-minimal permutations

We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.

Enumeration of standard barely set-valued tableaux of shifted shapes

A standard barely set-valued tableau of shape $λ$ is a filling of the Young diagram $λ$ with integers $1,2,\dots,|λ|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to the coincidental down-degree expectations (CDE) of lower intervals in Young's lattice. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shape. We show how it can be used to recover two formulas, or…

三角划分：枚举、结构与生成

Triangular partitions: enumeration, structure, and generation

本文研究了三角划分，给出了其新刻画、Möbius 函数公式、适合正方形的计数公式，并提出了高效生成算法，可计算大小至 10^5 的三角划分数量。

Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions

Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.

Enumeration of row-increasing tableaux of two-row skew shapes

In this paper, we firstly extend a result of Bonin, Shapiro and Simion by giving the distribution of the major index over generalized Schröder paths. Then by providing a bijection between generalized Schröder paths and row-increasing tableaux of skew shapes with two rows, we obtain the distribution of the major index and the amajor index over these tableaux, which extends a result of Du, Fan and Zhao. We also generalize a result of Pechenik and give the distribution of the major index over increasing tableaux of skew shapes with two rows. Especially, a bijection from row-increasing tableaux wi…

模4下偶数维分拆的计数

Enumeration of Even Dimensional Partitions modulo 4

本文借助2核塔理论，将奇维分拆的计数结论拓展至模4余2维的分拆，给出无连续二进制1的n对应的显式计数结果与通用递推公式。

Algebraic area enumeration of random walks on the honeycomb lattice

We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that the generating function of closed walks maps to the grand partition function of a system of particles with exclusion statistics of order $g=2$ and an appropriate spectrum, along the lines of a connection previously established by two of the authors. Reinterpreting the results in terms of the standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) a…

斐波那契数和卢卡斯数来源于轮图两组件生成森林

Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs

本文通过构造性双射和分析方法，推导出轮图两组件生成森林与斐波那契数、卢卡斯数的关系。

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).

A Determinantal Formula for Catalan Tableaux and TASEP Probabilities

We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with $α$'s and $β$'s that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with …

Investigating independent subsets of graphs, with Mathematica

With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to highlight the correspondence with other combinatorial objects with the same cardinality. Then we will study the structures obtained by ordering properly independent subsets of paths and cycles. We will approach some enumeration problems on the resulting partially ordered sets, putting in evidence the correspondences with structures known as Fibonacci and Lucas Cub…

Differential posets and Smith normal forms

We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.

Dual Filtered Graphs

We define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. The key formula in the definition is DU-UD= D + I. Our major examples are K-theoretic analogues of Young's lattice, of shifted Young's lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Mobius construction. The Pieri construc…

Set-Valued Tableaux & Generalized Catalan Numbers

Standard set-valued Young tableaux are a generalization of standard Young tableaux in which cells may contain more than one integer, with the added conditions that every integer at position $(i,j)$ must be smaller than every integer at positions $(i,j+1)$ and $(i+1,j)$. This paper explores the combinatorics of standard set-valued Young tableaux with two-rows, and how those tableaux may be used to provide new combinatorial interpretations of generalized Catalan numbers. New combinatorial interpretations are provided for the two-parameter Fuss-Catalan numbers (Raney numbers), the rational Catala…

Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young t…

Combinatorics of Triangular Partitions

The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, wi…

Okada 代数与幺半群的图模型

Diagram model for the Okada algebra and monoid

本文使用带标记的 Temperley-Lieb 弧图实现 Okada 代数及相关幺半群，证明其维数为 $n!$，并构造了该代数的细胞基。

Hall-Littlewood多项式、仿射Schubert级数与格枚举

Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

本文引入Hall-Littlewood-Schubert级数，通过变量替换解决p-adic格枚举等问题，提供Hecke级数新公式，并作为Stanley-Reisner环Hilbert级数的q-模拟，推广Littlewood恒等式。

Duality of graded graphs through operads

Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use operads, that algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric …

Young 图中的台球轨道：中缀链、自行车空间与多米诺骨牌平铺

Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings

本文研究 Young 图中对角台球轨道的计数，将其与中缀链、二进制自行车空间及多米诺骨牌平铺联系起来，给出精确公式、奇偶性限制及渐近下界。

Boxed Plane Partitions as an Exactly Solvable Boson Model

Plane partitions naturally appear in many problems of statistical physics and quantum field theory, for instance, in the theory of faceted crystals and of topological strings on Calabi-Yau threefolds. In this paper a connection is made between the exactly solvable model with the boson dynamical variables and a problem of enumeration of boxed plane partitions - three dimensional Young diagrams placed into a box of a finite size. The correlation functions of the boson model may be considered as the generating functionals of the Young diagrams with the fixed heights of its certain columns. The ev…

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find …

Osculating Paths and Oscillating Tableaux

The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right boundaries of a rectangle in the square lattice, each path can take only unit steps rightwards or upwards, and two different paths are permitted to share lattice points, but not to cross or share lattice edges. Such paths correspond to configurations of the six-vertex model of statistical mechanics with appropriate boundary conditions, and they include cases w…

$k$-配对：划分配对定理的推广

$k$-Pairing: A Generalization of the Partition Pairing Theorems

本文推广Andrews-Dastidar划分配对理论至$k$-配对，通过两个保权双射给出配对指标-宽度分布与负秩枚举的组合解释，并扩展至Young图元组等价类与平面划分。

图示Okada幺半群与Okada代数的胞腔性

Diagrammatic Okada monoid and cellularity of the Okada algebra

本文利用带标记非交叉弧图实现Okada代数与Okada幺半群，证明其维数、正则性等性质，构造其胞腔基并提出相关猜想，推进对称群与Okada代数的类比研究。

Arctic curves phenomena for bounded lecture hall Tableaux

Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded tableaux thanks to two other combinatorial models: non intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octogons. We use the tangent method to investigate the arctic curve i…