Related papers: Bruhat intervals that are large hypercubes

We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in…

The higher Bruhat orders $\mathcal{B}(n,k)$ were introduced by Manin-Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected $\mathcal{B}(n,k)$ to oriented matroids. In this paper, we…

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the…

We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the…

Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and…

Dasgupta recently introduced a cost function for the hierarchical clustering of a set of points given pairwise similarities between them. He showed that this function is NP-hard to optimize, but a top-down recursive partitioning heuristic…

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of…

We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals…

Let $\mathrm{Int}(n)$ denote the set of nonempty left weak Bruhat intervals in the symmetric group $\mathfrak{S}_n$. We investigate the equivalence relation $\overset{D}{\simeq}$ on $\mathrm{Int}(n)$, where $I \overset{D}{\simeq} J$ if and…

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this…

We investigate the unsupervised node classification problem on random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities. In this model, edges appear independently with probabilities…

We study the values of the M\"obius function $\mu$ of intervals in the containment poset of permutations. We construct a sequence of permutations $\pi_n$ of size $2n-2$ for which $\mu(1,\pi_n)$ is given by a polynomial in $n$ of degree 7.…

Calculation of near-neighbor interactions among high dimensional, irregularly distributed data points is a fundamental task to many graph-based or kernel-based machine learning algorithms and applications. Such calculations, involving…

Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation…

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known…

On May 14, 2025, DeepMind announced that AlphaEvolve, a large language model applied to a set of mathematical problems, had matched or exceeded the best known bounds on several problems. In the case of the sum and difference of sets…

We prove strong consistency of spectral clustering under the degree-corrected hypergraph stochastic block model in the sparse regime where the maximum expected hyperdegree is as small as $\Omega(\log n)$ with $n$ denoting the number of…

We revisit $R$-polynomials with introducing the new idea ``shifted $R$-polynomials" (or Bruhat weight) for all Bruhat intervals in finite Coxeter groups. Then, we apply these polynomials to weighted counting of Bruhat paths. Further, we…

We give new rounding schemes for SDP relaxations for the problems of maximizing cubic polynomials over the unit sphere and the $n$-dimensional hypercube. In both cases, the resulting algorithms yield a $O(\sqrt{n/k})$ multiplicative…

The binary symmetric stochastic block model deals with a random graph of $n$ vertices partitioned into two equal-sized clusters, such that each pair of vertices is connected independently with probability $p$ within clusters and $q$ across…