Bijective enumeration of rook walks

Bijective enumeration of permutations starting with a longest increasing subsequence

We prove a formula for the number of permutations in $S_n$ such that their first $n-k$ entries are increasing and their longest increasing subsequence has length $n-k$. This formula first appeared as a consequence of character polynomial calculations in recent work of Adriano Garsia and Alain Goupil. We give two `elementary' bijective proofs of this result and of its $q$-analogue, one proof using the RSK correspondence and one only permutations.

Bijective enumerations of $Γ$-free 0-1 matrices

We construct a new bijection between the set of $n\times k$ $0$-$1$ matrices with no three $1$'s forming a $Γ$ configuration and the set of $(n,k)$-Callan sequences, a simple structure counted by poly-Bernoulli numbers. We give two applications of this result: We derive the generating function of $Γ$-free matrices, and we give a new bijective proof for an elegant result of Aval et al. that states that the number of complete non-ambiguous forests with $n$ leaves is equal to the number of pairs of permutations of $\{1,\dots,n\}$ with no common rise.

Bijection between Increasing Binary Trees and Rook Placements on Double Staircases

In this paper, we shall construct a bijection between rook placements on double staircases (introduced by Josuat-Vergès in 2017) and increasing binary trees. We introduce two subclasses of rook placements on double staircases, which we call left and right-aligned rook placements. We show that their enumeration, while keeping track of a certain statistic, gives the $γ$-vectors of the Eulerian polynomials. We conclude with a discussion on a different bijection that fits in very well with our main bijection, and another discussion on generalising our main bijection. Our main bijection is a specia…

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…

Bijective enumerations for symmetrized poly-Bernoulli polynomials

Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.

An elementary approach to solve recursions relative to the enumeration of S-Motzkin paths

S-Motzkin paths (bijective to ternary trees) and partial version of them are calculated using only elementary methods from linear algebra.

排列深度与超位数的双射计数及符号不平衡

Bijective Enumeration and Sign-Imbalance for Permutation Depth and Excedances

本研究提出Biane双射的简化变体，通过带权3色Motzkin路径构造排列对合，证明了深度与超位数的奇偶计数性质，并给出按二者细化的排列及错位排列符号不平衡结果。

Bijective enumeration of some colored permutations given by the product of two long cycles

Let $γ_n$ be the permutation on $n$ symbols defined by $γ_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $β$ of $n$ in which the numbers from 1 to $n$ are colored with $p$ colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that $γ_n β^{-1}$ is a long cycle is given by the very simple ratio $\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportio…

一般栈的双射枚举

Bijective enumeration of general stacks

本文解决了蛋白质接触图中具有任意顶点度数上限的一般栈的计数难题，通过建立一般栈与$m$-正则$Λ$-规避$DLU$路径的双射，利用模式规避格路径理论推导出了统一的生成函数方程系统，并将已有RNA和蛋白质枚举结果作为特例统一其中。

Some simple bijections involving lattice walks and ballot sequences

In this note we observe that a bijection related to Littelmann's root operators (for type $A_1$) transparently explains the well known enumeration by length of walks on $\N$ (left factors of Dyck paths), as well as some other enumerative coincidences. We indicate a relation with bijective solutions of Bertrand's ballot problem: those can be mechanically transformed into bijective proofs of the mentioned enumeration formula.

非退化 $2 \times (k+1) \times k$ 超矩阵的计数

Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

本文研究了有限域上非退化 $2 \times (k+1) \times k$ 超矩阵的计数问题，提出并证明了乘积公式，关联到三维车棋放置的计数。

Enumerative and bijective aspects of combinatorial maps: generalization, unification and application (PhD thesis)

This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumer…

Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working…

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $λ= 1^{e_1}2^{e_2} \ldots$ of a tree on the vertex set $\set{1,2,\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corr…

Bijective link between Chapoton's new intervals and bipartite planar maps

In 2006, Chapoton defined a class of Tamari intervals called "new intervals" in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called "degree tree". Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.

A new bijective proof of Babson and Steingr\'ımsson's conjecture

Babson and Steingr\'ımsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $π$, let $des(π)$ denote the descent number of $π$ and $maj(π)$ denote the major index of $π$. Babson and Steingr\'ımsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-en…

A semi-bijective algorithm for saturated extended 2-regular simple stacks

Combinatorics of biopolymer structures, especially enumeration of various RNA secondary structures and protein contact maps, is of significant interest for communities of both combinatorics and computational biology. However, most of the previous combinatorial enumeration results for these structures are presented in terms of generating functions, and few are explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas for a particular class of biologically relevant structures, say, saturated 2-regular simple stacks, whose configuration is related to protein fol…

Narayana polynomials and some generalizations

In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides combinatorial proofs for a bijective problem in Stanley's collection "Bijective Proof Problems", a new formula for the Narayana polynomials as well as a new expression for the Harer-Zagier formula enumerating unicellular maps, in a unified way. Furthermore, we identify a class of plane trees, whose enumeration is closely connected to the Schröder numbers. Many other b…

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。

雅可比排列若干猜想的双射证明

Bijective proofs of several conjectures on Jacobi permutations

本文通过建立涉及多种组合结构的显式双射，证明了Henke等人提出的关于雅可比排列统计量分布的三个猜想，还得到了雅可比排列的三元指数生成函数闭式公式。

与交替符号矩阵相关的新双射证明

New bijective proofs pertaining to alternating sign matrices

本文针对 ASM-DPP 双射问题，利用相容性概念构造交替符号矩阵与移位 Gelfand-Tsetlin 模式间更自然的带符号双射，并为含 n+3 个统计量的 ASM 推广的精细化枚举提供双射证明。

Smirnov词与完全多部图中哈密顿圈的双射

Bijections Between Smirnov Words and Hamiltonian Cycles in Complete Multipartite Graphs

建立平衡字母重数的Smirnov词与完全m部图K_{n,n,...,n}中哈密顿路径的双射，导出哈密顿圈数的闭式容斥公式，并推广到非均匀情形K_{n1,n2,...,nm}，给出渐近分析。