Related papers: Counting Baxter Matrices
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
We provide a number of algorithmic results for the following family of problems: For a given binary m\times n matrix A and integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an…
For the calculation of Springer numbers (of root systems) of type $B_n$ and $D_n$, Arnold introduced a signed analogue of alternating permutations, called $\beta_n$-snakes, and derived recurrence relations for enumerating the…
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like…
We prove that the number of oscillating tableaux of length $n$ with at most $k$ columns, starting at $\emptyset$ and ending at the one-column shape $(1^m)$, is equal to the number of standard Young tableaux of size~$n$ with $m$ columns of…
The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a…
We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass…
In this thesis, we apply the stack sorting operator to $r$-permutations and construct the functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations counted by descents by using a factorization similar…
Baxter permutations are known to be in bijection with a wide number of combinatorial objects. Previously, it was shown that each of these objects had a natural involution which was carried equivariantly by the known bijections, and the…
In this article, we give a polynomial algorithm to decide whether a given permutation $\sigma$ is sortable with two stacks in series. This is indeed a longstanding open problem which was first introduced by Knuth. He introduced the stack…
The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix,…
We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…
This paper presents two alternative approaches for counting the number of two-row weakly increasing matrices, which are $2\times n$ matrices whose entries are integers from $1$ to $k$ and are weakly increasing along all rows and columns,…
We recall the concept of Baxterisation of an R-matrix, or of a monodromy matrix, which corresponds to build, from one point in the $ R$-matrix parameter space, the algebraic variety where the spectral parameter(s) live. We show that the…
A permutation $\pi: [n] \rightarrow [n]$ is a Baxter permutation if and only if it does not contain either of the patterns $2-41-3$ and $3-14-2$. Baxter permutations are one of the most widely studied subclasses of general permutation due…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…
A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…
Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider…
The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of…