Related papers: Higher Lie characters and cyclic descent extension…

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of…

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the…

We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating…

We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first charac- terize some functions having linear translators, based on which several families of…

Pattern avoidance for permutations has been extensively studied, and has been generalized to vincular patterns, where certain elements can be required to be adjacent. In addition, cyclic permutations, i.e., permutations written in a circle…

We examine the number of cycles of length k in a permutation, as a function on the symmetric group. We write it explicitly as a combination of characters of irreducible representations. This allows to study formation of long cycles in the…

We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a…

We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We show that the $k$-th moments of these statistics on…

This paper investigates the theory of lattices, focusing on extending lattices relative to abstract classes, modular lattices, and torsion lattices. Definitions of type-1 and type-2 extending lattices are provided, along with their weakly…

We determine when a permutation with cycle type $\mu$ admits a non-zero invariant vector in the irreducible representation $V_\lambda$ of the symmetric group. We find that a majority of pairs $(\lambda,\mu)$ have this property, with only a…

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on…

We study some extension of a discrete Heisenberg group coming from the theory of loop-groups and find invariants of conjugacy classes in this group. In some cases including the case of the integer Heisenberg group we make these invariants…

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle…

Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $\mathrm{Perm}$, the group of all recursive permutations $\mathcal{G}$. Multiple equivalent…

In 2010, B. Rhoades proved that promotion together with the fake-degree polynomial associated with rectangular standard Young tableaux give an instance of the cyclic sieving phenomenon. We extend this result to all skew standard Young…

In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence--modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two…

Let $\beta$ be any permutation on $n$ symbols and let $c(k, \beta)$ be the number of permutations that $k$-commute with $\beta$. The cycle type of a permutation $\beta$ is a vector $(c_1, \dots, c_n)$ such that $\beta$ has exactly $c_i$…

In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In…

One takes advantage of some basic properties of every homotopic $\lambda$-model (e.g.\ extensional Kan complex) to explore the higher $\beta\eta$-conversions, which would correspond to proofs of equality between terms of a theory of…

We prove a moving lemma for the additive and ordinary higher Chow groups of relative $0$-cycles of regular semi-local $k$-schemes essentially of finite type over an infinite perfect field. From this, we show that the cycle classes can be…