Related papers: Sorting probability for large Young diagrams
Felsner and Reuter introduced the linear extension diameter of a partially ordered set $\mathbf{P}$, denoted $\mbox{led}(\mathbf{P})$, as the maximum distance between two linear extensions of $\mathbf{P}$, where distance is defined to be…
A grid poset -- or grid for short -- is a product of chains. We ask, what does a random linear extension of a grid look like? In particular, we show that the average "jump number," i.e., the number of times that two consecutive elements in…
Although an input distribution may not majorize a target distribution, it may majorize a distribution which is close to the target. Here we introduce a notion of approximate majorization. For any distribution, and given a distance $\delta$,…
In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…
In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by $\pi/4$). We also give an…
A sorting network is a shortest path from 12..n to n..21 in the Cayley graph of the symmetric group S(n) generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove…
The pop-stack-sorting process is a variation of the stack-sort process. We consider a deterministic version of this process, and provide a new lower bound of $\frac{3}{5}n$ for the number of sorts to fully sort a uniformly randomly chosen…
Kim, Kim, and Neggers (2019) defined probability functions on a poset, by listing some very natural conditions that a function \(\pi: P \times P \to [0,1]\) should satisfy in order to capture the intuition of "the likelihood that \(a\)…
We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes…
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear…
In this paper we consider the density of maximal order elements in $\mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $\mathbb{F}_q$ of size…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the…
Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $\lambda$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $\lambda$ has a unique…
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it.…
The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade)…
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…
On the lattice $\widetilde{\mathbb Z}^2_+:={(x,y)\in \mathbb Z \times \mathbb Z_+\colon x+y \text{is even}}$ we consider the following oriented (northwest-northeast) site percolation: the lines $H_i:={(x,y)\in \widetilde {\mathbb Z}^2_+…
One of possible cryptomorphic definitions of a partially ordered set (= a poset) $P$ on a non-empty finite basic set $N$ is in terms of the set ${\cal L}(P)$ of all its linear extensions, that is, in terms of the set of total orders of $N$…
A lower bound on the probability $P(0<X<\delta)$ for all real $\delta>0$ and all random variables $X$ with log-concave p.d.f.'s such that $EX=0$ and $EX^2=1$ is obtained.
A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the…