Related papers: Sorting probability for large Young diagrams
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is…
The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behavior. In this paper, for a higher order Markov chain, we establish some properties related to its exact limiting…
Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of…
A striking result of [Acharya et al. 2017] showed that to estimate symmetric properties of discrete distributions, plugging in the distribution that maximizes the likelihood of observed multiset of frequencies, also known as the profile…
The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $\pi$…
We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that…
We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM},…
Sorting has a natural generalization where the input consists of: (1) a ground set $X$ of size $n$, (2) a partial oracle $O_P$ specifying some fixed partial order $P$ on $X$ and (3) a linear oracle $O_L$ specifying a linear order $L$ that…
We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there…
In this paper, we introduce a convergence notion for ordered selections. Our convergence notion is based on subpermutation densities and convergences of the marginal distributions. A particular case of this convergence is the well-known…
In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…
We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young…
A sorting network is a shortest path from $12\dots n$ to $n\dots 21$ in the Cayley graph of the symmetric group $\mathfrak S_n$ spanned by adjacent transpositions. The paper computes the edge local limit of the uniformly random sorting…
A poset $\mathbf{P} = (X,\preceq)$ is {\em $m$-partite} if $X$ has a partition $X = X_1 \cup ... \cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In…
We investigate the poset (P(X),\subset), where P(X) is the set of isomorphic suborders of a countable ultrahomogeneous partial order X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in…
We show that the persistence probability $P(t,L)$, in a coarsening system of linear size $L$ at a time $t$, has the finite size scaling form $P(t,L)\sim L^{-z\theta}f(\frac{t}{L^{z}})$ where $\theta$ is the persistence exponent and $z$ is…
Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced…
Large Language Models are transformer neural networks which are trained to produce a probability distribution on the possible next words to given texts in a corpus, in such a way that the most likely word predicted is the actual word in the…
A general principle is advanced allowing the classification of nonunique solutions to nonlinear evolution equations, corresponding to different spatio-temporal patterns. This is done by defining the probability distribution of patterns,…
We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert…