115 problems

Duval–Klivans–Martin conjecture. For every graph $G$, the Pak–Stanley labeling is a surjection from the regions of the $G$-Shi arrangement onto the $G$-parking functions.

Rational shuffle generating-function conjecture. The polynomial

Rational parking function symmetry conjecture. The polynomial $PF_{m,n}(q,t)$ is symmetric in $q$ and $t$, namely

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…

Gorsky–Mazin–Vazirani conjecture. The map $SP:{}^m\widetilde{S_n}\to PF_n^m$ is bijective for all coprime $m$ and $n$.

Let $(m,n)$ be a coprime pair, let $Q_{m,n}$ be the operator recursively defined from the splitting of $(m,n)$, and let $\operatorname{PF}_{m,n}$ denote the corresponding set of…

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

Bergeron–Shapiro level-one conjecture. The function $b$ satisfies

Bergeron–Shapiro conjecture. There exists an integer-valued function $b=b(l)$ such that the number of admissible pairs $(k,l)$ satisfying

Haglund–Loehr conjecture. The coefficient of $q^at^b$ in $R_n(q,t)$ equals

Freeness conjecture. Both Lie algebras ${\bf LPQ}^*$ and ${\bf LPQ}$ are free on generators whose degree generating function is the series above. This is an analogue of the corresp…

Armstrong–Reiner–Rhoades' character conjecture. The character of $\operatorname{PF}$ sends $(c^k,w)$ to

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…

Kostka-weighted formula. The polynomial $C^f_{m,n}(q,t)$ coincides with the bigraded polynomial arising from either the mixed Hodge filtration or the perverse filtration of the Hit…

Unimodality conjecture. The sequence $(a_\ell)_{\ell=0}^{n-1}$ is unimodal: there exists an integer $c\in\{0,1,\dots,n-1\}$ such that

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…

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…

Wedge-of-spheres conjecture.

Let $e_n$ be the elementary symmetric function, let $\Delta'_{e_{k-1}}$ denote the modified Macdonald eigenoperator indexed by $e_{k-1}$, and let…

Let a parking function be chosen uniformly at random, and let its area be the associated area statistic. Bóna's conjecture. The area of a uniformly random parking function has a lo…

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…