On convex sets associated with permutations, cuts, and certain metrics

Convex cones and SAGBI bases of permutation invariants

Let G be a permutation group acting on {1,...,n}, and < be any admissible term order on the polynomial ring K[x_1,...,x_n]. We prove that the invariant ring K[x_1,...,x_n]^G of G has a finite SAGBI basis if, and only if, G is generated by reflections.

Convex obstacle numbers of outerplanar graphs and bipartite permutation graphs

The disjoint convex obstacle number of a graph G is the smallest number h such that there is a set of h pairwise disjoint convex polygons (obstacles) and a set of n points in the plane (corresponding to V(G)) so that a vertex pair uv is an edge if and only if the corresponding segment uv does not meet any obstacle. We show that the disjoint convex obstacle number of an outerplanar graph is always at most 5, and of a bipartite permutation graph at most 4. The former answers a question raised by Alpert, Koch, and Laison. We complement the upper bound for outerplanar graphs with the lower bound o…

A bijection between triangulations and $312$-avoiding permutations

We define a bijection between triangulations of a convex polygon and $312$-avoiding permutations through the process of "ear-clipping". This bijection is then used to obtain a bijection between polygon dissections and a certain class of $312$-avoiding permutations.

Logical limit laws for layered permutations and related structures

We show that several classes of ordered structures (namely, convex linear orders, layered permutations, and compositions) admit first-order logical limit laws.

On permutation polytopes

A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in dimensions 2,3,4, and the corresponding permutation groups up to a suitable notion of equivalence. We also provide a list of combinatorial types of possibly occuring faces of permutation polytopes up to dimension four.

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polyno…

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。

Permutation Polytopes of Cyclic Groups

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.

On the exhaustive generation of convex permutominoes

A permutomino of size n is a polyomino determined by a pair of permutations of size n+1, such that they differ in each position. In this paper, after recalling some enumerative results about permutominoes, we give a first algorithm for the exhaustive generation of a particular class of permutominoes, the convex permutominoes, proving that its cost is proportional to the number of generated objects.

Positroid Catalan numbers

Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky hom…

A型Demazure特征标（关键多项式）的表集合凸性，抛物Catalan数

Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

本文证明A型Demazure特征标对应的半标准杨表集合的凸性等价于R-312-避免条件，并定义R-抛物Catalan数，为后续深化Reiner-Shimozono及Postnikov-Stanley结果奠定基础。

Convex $(0,1)$-Matrices and Their Epitopes

We investigate $(0,1)$-matrices that are {\em convex}, which means that the ones are consecutive in every row and column. These matrices occur in discrete tomography. The notion of ranked essential sets, known for permutation matrices, is extended to convex sets. We show a number of results for the class $\mc{C}(R,S)$ of convex matrices with given row and column sum vectors $R$ and $S$. Also, it is shown that the ranked essential set uniquely determines a matrix in $\mc{C}(R,S)$.

Largest minimal inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where $1 \le i \textless{} j \le n$, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and…

Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters

Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number…

Affine maps between quadratic assignment polytopes and subgraph isomorphism polytopes

We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P(…

由Baxter数产生的解析性质

Analytic properties arising from the Baxter numbers

本文研究Baxter数的解析性质，基于Dilks的$(q,t)$-Hoggatt和框架证明了Baxter置换下降多项式具有交错零点，并通过$P$-递归序列的渐近展开证明了Baxter数的渐近$r$-对数凸性，特别确认了其$2$-对数凸性。

Constraint metric approximations and equations in groups

We introduce notions of a constraint metric approximation and of a constraint stability of a metric approximation. This is done in the language of group equations with coefficients. We give an example of a group which is not constraintly sofic. In building it, we find a sofic representation of free group with trivial commutant among extreme points of the convex structure on the space of sofic representations. We consider the centralizer equation in permutations as an instance of this new general setting. We characterize permutations $p\in S_k$ whose centralizer is stable in permutations with r…

A property of the Birkhoff polytope

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices in $\mathbb{R}^{n\times n}$. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group $G\leq \operatorname{GL}(d,\mathbb{R})$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erdős-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric…

Convex Rank Tests and Semigraphoids

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The…