Related papers: From Parking Functions to Gelfand Pairs

A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure…

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic…

Let $W$ be an irreducible real reflection group. Armstrong, Reiner, and the author presented a model for parking functions attached to W and made three increasingly strong conjectures about these objects. The author generalized these…

We give a recursive definition of generalized parking function that allows us to view them as a species. From there we compute a non-commutative characteristic of the generalized parking function module, and deduce some enumeration formulas…

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their…

For any integers $1\leq k\leq n$, we introduce a new family of parking functions called $k$-vacillating parking functions of length $n$. The parking rule for $k$-vacillating parking functions allows a car with preference $p$ to park in the…

The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product…

Given a compact Gelfand pair (G,K) and a locally compact group L, we characterize the class P_K^\sharp(G,L) of continuous positive definite functions f:G\times L\to \C which are bi-invariant in the G-variable with respect to K. The…

A strong Gelfand pair (G,H) is a group G together with a subgroup H such that every irreducible character of H induces a multiplicity-free character of G. We classify the strong Gelfand pairs of the special linear groups SL(2, p) where p is…

We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of…

Let $G$ be a finite group acting on a finite set $X$. This group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is projected to orbits. We…

We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and {\L}ukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the…

We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions.…

The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in…

Let S be the group of finite permutations of the naturals 1,2,... The subject of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands for the product of two copies of S while K is the diagonal subgroup in G. The…

Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$ vertices are visited before backtracking. Let $R$ be the set of trees with this property.…

Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the…

We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and…

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution on…

This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space…