Related papers: Uniquely-Wilf classes

We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a…

We denote by Conc(L) the semilattice of all finitely generated congruences of a lattice L. For varieties (i.e., equational classes) V and W of lattices such that V is contained neither in W nor its dual, and such that every simple member of…

Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on…

Let $P(n)$ be the probability that two independent, uniformly random permutations of $[n]$ have the same order, and let $K(n)$ be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove…

Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of…

Given a permutation w, we show that the number of repeated letters in a reduced decomposition of w is always less than or equal to the number of 321- and 3412-patterns appearing in w. Moreover, we prove bijectively that the two quantities…

Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1,…

Let Q be a connected directed quiver with n vertices. We show that Q is representation-infinite if and only if there do exist n isomorphism classes of exceptional modules of some fixed length at least 2.

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further…

Assuming that the two integrals in the Change of Variable Formula for the Riemann integral on the real line are finite, one can rightfully ask if they have equal value. We give a positive answer to this question. The proof is very easy to…

Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of…

We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a…

It is shown that for the conjugation action of the symmetric group $S_n,$ when $n=6$ or $n\geq 8,$ all $S_n$-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition $\lambda$ of $n$ with at least…

The classes of tree permutations and forest permutations were defined by Acan and Hitczenko (2016). We study random permutations of a given length from these classes, and in particular the number of occurrences of a fixed pattern in one of…

If every point of a unital is fixed by a non-trivial translation and at least one translation has order two then the unital is classical (i.e., hermitian).

In this article, we describe an algorithm to determine whether a permutation class C given by a finite basis B of excluded patterns contains a finite number of simple permutations. This is a continuation of the work initiated in [Brignall,…

Partially ordered patterns (POPs) generalize classical permutation patterns and have been extensively studied in the contexts of permutations, words, compositions, and partitions. Burstein, Han, Kitaev, and Zhang established the…

A permutation $\pi \in \mathbb{S}_n$ is $k$-balanced if every permutation of order $k$ occurs in $\pi$ equally often, through order-isomorphism. In this paper, we explicitly construct $k$-balanced permutations for $k \le 3$, and every $n$…

Multidimensional permutations, or $d$-permutations, are represented by their diagrams on $[n]^d$ such that there exists exactly one point per hyperplane $x_i$ that satisfies $x_i= j$ for $i \in [d]$ and $j \in [n]$. Bonichon and Morel…