50 problems

Let $\V$ be the poset appearing in the product $\V\times [k]$. Consider piecewise-linear rowmotion on the order polytope of $\V\times [k]$, and rowmotion on the order ideals of…

Let $W$ be a finite crystallographic Coxeter group with Coxeter number $h$, let $J(\Phi^+(W))$ be the order ideals of its positive root poset, and let $d_1,d_2,\ldots,d_n$ be the d…

Alt $\nu$-Tamari homometry conjecture. The down-degree statistic is homometric for rowmotion on alt $\nu$-Tamari lattices $\mathsf{Tam}_{\delta}(\nu)$ and is independent of the inc…

Let $[m]\times[n]$ be the product of two chains, and let an interval-closed set mean a subset that is interval-closed in this poset. For $x\in[m]\times[n]$, define the signed cardi…

Let $W$ be a finite Weyl group with positive-root poset $\Phi^+$, rank $r$, Coxeter number $h$, and longest element $w_0\in W$. Let $\operatorname{NN}(W)=\mathcal{A}(\Phi^+)$ be th…

Let $V(n)$ be the poset in the source, let $r(P)$ denote the rank of a poset $P$, let $\operatorname{Row}$ denote rowmotion, and let $\delta$ denote the involution used in the sour…

Let $P$ be a finite graded poset of rank $r$ for which the roots of $\aΩ_P(m)$ are all integers or half-integers, and in the latter case possessing an order two automorphism. Let…

Orbit-structure conjecture. The orbit structure of the alt $\nu$-Tamari lattice $\mathsf{Tam}_{\delta}(\nu)$ is independent of the increment vector $\delta$.

Let $R$ be a finite poset whose incidence algebra over a field of characteristic $0$ is Auslander-regular, and suppose that $R$ is connected. A finite poset is echelon-independent…

Let $P=[m]\times[n]$ be the product of two chains, and let $\mathcal{IC}(P)$ denote its interval-closed sets. Under rowmotion on $\mathcal{IC}(P)$, consider the statistic that assi…

Fix a positive integer $m$. Let $r(n)$ be the fraction of interval-closed sets of $[m]\times[n]$ that belong to rowmotion orbits of size $m+n+1$. Typical-orbit conjecture. For ever…

Let $a$ be a positive integer, let $t$ be odd, and let $F=\breve{F}(a^t)$ be the uniform self-dual fence with $t$ segments. For $i\in\{1,\ldots,t-1\}$, let $s_i$ and $s_{t-i}$ be t…

Let $a$ be a positive integer, let $t$ be odd, and let $F=\breve{F}(a^t)$ be the uniform self-dual fence with $t$ segments. For each peak $p$ and valley $v$ of $F$, let $\chi_p$ an…

Let $a$ be a positive integer, let $t$ be odd, and let $F=\breve{F}(a^t)$ be the uniform self-dual fence with $t$ segments. Write $n=\#F$ for its number of elements, and let…

Let $F$ be a fence, and let $\hat{\chi}$ denote its order ideal cardinality statistic. Elizalde et al.'s homomesy conjecture. For certain fences, $\hat{\chi}$ is homomesic under ro…

Let $T_{r,s}$ be the trapezoidal poset with parameters $r$ and $s$, and let birational rowmotion act on its labelings. Grinberg–Roby's periodicity conjecture. Birational rowmotion…

Maximal-minus-minimal homomesy conjecture. The number of maximal elements minus the number of minimal elements is $0$-mesic under rowmotion on $\mathcal{IC}(P)$ for…

Asymptotic homomesy conjecture.

Thomas–Williams' biCambrian orbit conjecture. The orbit structure of rowmotion on $B(W,c)$ is the same as the orbit structure of rowmotion on $\mathcal J(P_W)$.

Rational Tamari down-degree homomesy conjecture. The statistic $\operatorname{ddeg}$ is homomesic for rowmotion, with average

Thomas–Williams' cyclic sieving conjecture. The triple

Thomas–Williams' conjecture. The operator

Let $V=\{a,b,c\}$ be the three-element poset with $a\lessdot b$ and $a\lessdot c$, let $\mathcal{A}^{\ell}(P)$ denote the set of $P$-partitions of height $\ell$, and let…

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{A}^2(P)$ denote the set of $P$-partitions of height at most $2$, and let $\chi_S$ be the indicator statistic of $S$. For antipodal elements…

Let $F(\beta)$ be a fence and let $L(\beta)$ be its lattice of lower order ideals. Suppose $k\ge 2$ and $\beta=(k-1,k,k,\ldots,k,k-1)$ has an odd number of parts. Under rowmotion o…