15 problems

Rational shuffle generating-function conjecture. The polynomial

Let $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{C}_w(n,k)$ denote the relevant collection of consistent subsets, and let $R\in \mathcal{C}_w(n,k)$. Define $G_R$ by including e…

For a positive integer $n$ and an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, consider the higher-Bruhat-type theorem stated immediately before this conjecture, including…

For an affine permutation $w\in \widetilde{\mathfrak{S}}_n$, let $\mathcal{G}(w)$ be the directed graph whose vertices are commutation classes of reduced words for $w$, with edges…

Let $I$, $N$, $k$, and the word set $(\pm I)^m$ be as in the affine plabic-fence construction. For a word $\mathbf w$, let $\mathbf w_1$ and $\mathbf w_2$ be the associated positiv…

Let $G(\mathbf w,f)$ and $G(\mathbf w',f')$ be affine plabic fences, with restricted move equivalence defined using the relevant affine plabic-fence moves. Affine plabic-fence clas…

Bounded affine permutation growth-rate conjecture. The proper growth rate exists and equals the Stanley–Wilf limit for every sum-indecomposable pattern $\tau$:

Affine Stanley–Wilf growth-rate conjecture. The proper growth rate

Let $w$ be an affine permutation with AMBC data … where $P$ and $Q$ are tabloids of shape $\lambda$, and let $r(w)$ denote the rotation of $w$. For a vector, call its dominant repr…

Pak–Stanley bijectivity conjecture. The map $\mathcal{PS}_m$ is a bijection from $\operatorname{Stab}(n,m)$ to the set of $(n,m)$-parking functions for all $m$.

Permutation formulation of the rational Shuffle Conjecture. The following equation holds:

Affine inverse-algorithm termination conjecture. For the resulting $U$, there exists $N$ such that for every $i\ge N$ and every $t\in\mathbb{N}$,

The combinatorial Hilbert-series symmetry conjecture. The series is symmetric in $q$ and $t$ for all $m$ and $n$:

The rational parking-function bijectivity conjecture. The map $\operatorname{\mathcal{PS}}$ is bijective for all relatively prime $m$ and $n$.

Enumeration conjecture. The following equalities hold: