18 problems

Let $D_n^{(r)}$ be the set of all $r$-fans of Dyck paths of length $2n$, and define … where $[m]_q=1+q+q^2+\cdots+q^{m-1}$. Let $C_{2n}$ be the cyclic group of order $2n$ acting on…

Promotion conjecture. Theorem 4.11 remains true for any tensor product

Let $P$ be a poset with $n$ elements and $m$ minimal elements. A labeling of $P$ is tangled if it requires $n-1$ applications of extended promotion to become a natural labeling. Ho…

Tangled–quasi-tangled inequality. The number of tangled labelings of $P$ is less than or equal to the number of quasi-tangled labelings of $P$.

Coloring probability conjecture.

Refined tangled-labeling bound. $P$ has at most

Maximum noninvertibility conjecture. For any poset $P$,

Defant–Kravitz's conjecture. $P$ has at most $(n-1)!$ tangled labelings.

Let $[n]=\{1,\ldots,n\}$, let $\beta$ be the restriction function on $[2]\times[b]$ defined by … and let $\mathcal{L}_{([2]\times[b])\times[\ell]}(R^{\beta})$ denote the correspond…

Let $V=\{a,b,c\}$ be the three-element poset with $a\lessdot b$ and $a\lessdot c$, let $\mathcal{L}_{V\times[\ell]}(R^q)$ denote the set of $V$-strict labelings with restriction fu…

Let $\mathcal{L}_{P\times[\ell]}(R^q)$ denote the set of labelings with the stated restriction function, and let $\xi$ be the statistic defined in the preceding formulation. Transl…

Let $\mathcal{L}_{P \times [\ell]}(R^q)$ denote the set of labelings subject to the indicated restriction function, let $\xi(f,(x_1,x_2),b)$ be the statistic defined by … and let…

Staircase flagged-tableaux cyclic sieving conjecture. The triple $(\mathrm{FT}(sc_n,b),\langle\operatorname*{Pro}\rangle,Cat_{\ell}(x))$ exhibits the cyclic sieving phenomenon.

Let $V(n)$ be the poset in the source, let $\mathcal{L}(V(n))$ denote its set of linear extensions, and let $\operatorname{Pro}$ denote promotion. Hopkins–Rubey's conjecture. For e…

Evacuation fixed-point conjecture. For all $n\geq 1$, the number of Kreweras words of length $3n$ satisfying $\operatorname{{Evac}}^{*}(w)=w$ is

A standard skew tableau has rows whose sizes weakly decrease from top to bottom; write $b$ for the number of rows and let $\stab$ denote its stabilization time, the least $k$ at wh…

Assume there is a bijection between generalized oscillating tableaux of length $k$ with $n$ parts, whose final component is $(m,\ldots,m)$ for some $m\in\mathbb{Z}$, and…

Periodicity conjecture. For every $T \in \operatorname{Inc}^q(3 \times n)$,