Cyclic derangements

经典Weyl群中错排上的带符号Mahonian多项式

Signed Mahonian Polynomials on Derangements in Classical Weyl Groups

本文推导了经典Weyl群（对称群、超八面体群和偶符号置换群）中错排上的（带符号）Mahonian多项式的公式，并应用这些公式得到了具有偶数长度的错排数量的枚举公式。

Cyclic sieving and orbit harmonics

Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in $\mathbb{C}^n$. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. We apply orbit harmonics to prove cyclic sieving results.

On a pattern avoidance condition for the wreath product of cyclic groups with symmetric groups

In this paper, we extend to a non-consecutive case, the study of the pattern matching condition on the wreath product of the cyclic group and the symmetric group initiated by the authors in a previous paper. The main focus of our paper is (colored) patterns of length 2, although a number of enumerative results for longer patterns are also presented. A new non-trivial bijective interpretation for the Catalan numbers is found, in terms of simultaneously bi-avoiding two patterns in a wreath product.

Enumerative properties of restricted words and compositions

In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count cyclically restricted and circular objects. This either fills gaps in the current literature by e.g. considering particular new patterns, or involves general progress, notably with locally restricted compositions over a finite group. We associate these compositions to walks on a covering graph whose structure is exploited to simplify asymptotic expressions. Specifica…

The cyclic sieving phenomenon: a survey

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many in…

On arithmetic partitions of Z_n

Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number of subsets of $\Z_n$ without certain separations. Chen, Wang, and Zhang then studied the problem of partitioning $\Z_n$ into arithmetical progressions of a given type under some technical conditions. In this paper, we improve on their main theorems by applying a convolution formula for cyclic multinomial coefficients due to Raney-Mohanty.