Related papers: Lattice Paths and Order-preserving Partial Transfo…

In this article we study decreasing and increasing factorisations of the cycle, which are decompositions of the cycle $(1~2\dots n)$ into a product of $n-1$ transpositions satisfying monotonicity conditions. We explicit a bijection between…

The states in the irreducible modules of the minimal models can be represented by infinite lattice paths arising from consideration of the corresponding RSOS statistical models. For the M(p,2p+1) models, a completely different path…

A poset $\bfp$ is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, $\ell(\bfp)$ of $\bfp$. We prove that if the vertex set $X$ of…

We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase…

We investigate possible problems with universality in lattice gauge theory where a mixed fundamental SU(2) and SO(3)-invariant gauge group is used: the (second order) finite temperature phase transition becomes involved with first order…

Consider an $m\times n$ table $T$ and latices paths $\nu_1,\ldots,\nu_k$ in $T$ such that each step $\nu_{i+1}-\nu_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is…

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$…

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L\"owner and the minus partial…

Let $P$ be a partially ordered set. We prove that if $n$ is sufficiently large, then there exists a packing $\mathcal{P}$ of copies of $P$ in the Boolean lattice $(2^{[n]},\subset)$ that covers almost every element of $2^{[n]}$:…

Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called {\it G-Motzkin paths} for short, that is lattice paths…

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points…

Let $\mathcal{ORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$, and for $1\leq r\leq n-1$, let $$\mathcal{ORD}(n,r) =\{\alpha \in…

Lattice paths called $\ell$-Schr\"oder paths are introduced. They are paths on the upper half-plane consisting of $\ell+2$ types of steps: $(i,\ell-i)$ for $i=0,\ldots,\ell$, and $(1,-1)$. Those paths generalize Schr\"oder paths and some…

We demonstrate mesoscopic transport through quantum states in quasi-1D lattices maintaining the combination of parity and time-reversal symmetries by controlling energy gain and loss. We investigate the phase diagram of the non-Hermitian…

In this paper, two new composition operations are defined among the order-preserving maps. They can act on order-preserving maps like the usual composition operation. They are coincide with the usual composition operation when the…

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the $x$-axis are enumerated by the generalized Catalan numbers, and we give a…

Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the…

The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In…

We focus on a family of subsets $(\F^p_n)_{p\geq 2}$ of Dyck paths of semilength $n$ that avoid the patterns $DUU$ and $D^{p+1}$, which are enumerated by the generalized Fibonacci numbers. We endow them with the partial order relation…

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.