56 problems

Let $n$ be a positive integer and let $j$ be a positive integer. A lucky spot is a parking spot occupied by a car whose preferred spot is that spot. Let $f_j(n)$ denote the polynom…

Let $n\geq 1$ and $k\in \{0,1,\dots,n(n-1)/2\}$. The quantity $\operatorname{upf}_{n,k}^{\operatorname{inv}}$ denotes the number of unit interval parking functions of length $n$ wi…

Let $m,d,n$ be nonnegative integers, and let $4\lambda\vdash m$ be a partition of $m$ of length $k$. For each $i$, let $4\mu_i(\lambda)$ be the number of parts of $4\lambda$ of siz…

Let $\mathcal{D}_{m,n}$ denote the set of rational Dyck paths with parameters $m,n\in\mathbb{Z}_{>0}$. For the parameter range $m<n$, find suitable run-type and return-type statist…

Let $\mathsf{PF}_n$ denote the parking function polytope. A triangulation of a lattice polytope is regular if it is induced by a lifting function, and unimodular if every simplex i…

Let $\mathsf{PF}_n$ be the set of parking functions of length $n$, and define the sum statistic by $\mathsf{\sum}(\pi)=a_1+a_2+\ldots+a_n$ for…

Wilmes' conjecture. For all $i\geq 1$, one has

The Type B non-decreasing parking functions are the elements of the submonoid of $\operatorname{NDPF}_{2N}$ generated by $\mu_i=\pi_i\pi_{2N-i}$ for $1\leq i\leq N$, with…

Let $\mathscr{H}_n^{(r)}(\mathbf{w};1,1,1)$ denote the specialization of the graded Frobenius characteristic of the higher diagonal harmonics at $q_1=q_2=1$. For a partition…

Let $(W,S)$ be an irreducible finite Coxeter system with Coxeter number $h$, let $\vec{c}$ be a Coxeter word in $S$, let $p>0$ be an integer coprime to $h$, and let $J\subseteq S$.…

Let ${\bf PQSym}^*$ denote the dual Hopf algebra of parking functions, realized as a dendriform trialgebra with operations induced from the dendriform trialgebra structure on…

Let $B_n$ be the signed symmetric group on $n$ letters, with inclusion $B_n\subset B_{n+1}$ given by choosing the elements of $B_{n+1}$ that fix the $(n+1)$st letter. For…

Let $I_n(q,t)$ and $\widetilde{I}_n(q,t)$ be the two $t$-analogues of the tree inversion enumerator, let $\mathrm{PF}(n)$ denote the set of parking functions of length $n$, and for…

Let $\alpha$ be a unit interval parking function, let $F$ denote the Foata transform, and let $\operatorname{spot}_\alpha$ denote the outcome permutation of $\alpha$. Foata transfo…

Wedge-of-spheres conjecture.

Let $m,n$ be positive integers, and let $\mathrm{mpf}_{m,n}(t)$ denote the number of $t$-metered $(m,n)$-parking functions. Monotonicity conjecture. If $t_1<t_2$, then … This conje…

A valleyless-tieless parking function of length $n$ is a parking function $(a_1,\ldots,a_n)$ with no index $1\leq i\leq n-1$ such that $a_i=a_{i+1}$ and no index $2\leq i\leq n-1$…

For $n\geq 1$, let $\mathcal{O}_{\mathrm{MVP}_{n}}$ denote the MVP outcome map on permutations of size $n$, and let $\mathrm{split}^{2,n-2}:=(n-1)n(n-2)(n-3)\cdots21$ for $n\geq3$.…

Let $\mathbf{y}=(y_1,y_2,y_3,y_4)\in\mathbb{N}^4$. A car-length vector is minimally invariant when it has the minimal invariance property defined in the paper. Four-car minimal-inv…

Let $\gamma$ and $\lambda$ be the parameters used to define the set $\operatorname{PF}^{\gamma}_{\lambda}$ of $\gamma$-parking functions, let $S$ be a subset of the area cells of…

Konvalinka–Tewari conjecture. The representation $\tau_{n,1}$ is isomorphic to $\sigma_n$. Furthermore, $\mathrm{Frob}(\tau_{n,1})$ is $h$-positive.

Let $n$ be a positive integer, let $abla$ be the nabla operator on symmetric functions, and let $e_n$ and $s_{\mu}$ denote the elementary and Schur symmetric functions, respective…

Goldschmidt and Przykucki's conjecture. The expected number of cars arriving at the root is

Let $n$ be a positive integer, and let a parking preference be represented by its associated lattice-path diagram, with corners defined as turns of the path. Suppose that there is…

Let $n\geq1$. For each ordered set partition $\Pi$ of $[n]$, let $\mathcal{B}_\Pi$ be the set of monomials constructed from the associated permutation, Grassmann monomial, and sche…