Related papers: On simultaneous conjugation of permutations

We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in…

Let G be a transitive group of permutations of a finite set X, and suppose that some element of G has at most two orbits on X. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and…

We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as "noncommutative endomorphisms" of a polynomial algebra. More explicitly…

We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas…

Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…

We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that…

We give a combinatorial proof that congruence permutability is prime in the lattice of interpretability types of varieties. Thereby, we settle a 1984 conjecture of Garcia and Taylor.

Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.

Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this…

A combinatorial group-theoretic hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This hypothesis has a component that can be expressed in terms of…

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central…

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct…

Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the…

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann…

Two permutations are similar if they have the same length and the same relative order. A collection of $r\ge2$ disjoint, similar subsequences of a permutation $\pi$ form $r$-twins in $\pi$. We study the longest guaranteed length of…

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the…

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is…

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively…

We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in…