Related papers: A new cyclic sieving phenomenon for Catalan object…
We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the…
The cyclic sieving phenomenon was introduced by Reiner, Stanton and White in 2004 as a generalization of Stembridge's $q=-1$ phenomenon. In a paper from 2008, Eu and Fu studied many occurrences of this phenomenon on the faces of the…
We will exhibit several instances of the cyclic sieving phenomenon involving statistics and involutions on the following combinatorial families of objects: permutations, set partitions, perfect matchings, D-permutations (and its…
It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed…
Let Z_m^k consist of the m^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A_k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order (k+1), so does Z_m^k. By bijectively…
We show an instance of the cyclic sieving phenomenon on annular noncrossing permutations with given cycle types. We define annular $q$-Kreweras numbers, annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are…
We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials, and more generally the $q$-Schr\"oder polynomials. By interpreting them as $\mathfrak{sl}_2$-characters, we show that the…
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…
An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a…
We prove an instance of the cyclic sieving phenomenon in non-crossing connected graphs, as conjectured by S.-P. Eu.
We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are established for binomial coefficients and the Ap\'{e}ry numbers, by means of a general…
We prove a cyclic sieving result for the set of $3 \times k$ packed increasing tableaux with maximum entry $m :=3+k$ under K-promotion. The "curiosity" is that the sieving polynomial arises from the $q$-hook formula for standard tableaux of…
Recently, a combinatorial model for torsion pairs in the cluster category of Dynkin type A_n was introduced, and used to derive an explicit formula for their number. In this article we determine the number of torsion pairs that are…
We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees…
We prove that the generalised non-crossing partitions associated to well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, respectively by Bessis and Reiner.…
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric…
We show that the cyclic sieving phenomenon of Reiner--Stanton--White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to…
In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving…
We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of…
Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash…