32 problems

Let $L$ be a modular lattice, and let $\sigma$ be a linear extension of $L$. For $x\in L$, write $\operatorname{Cov}_L^{\uparrow}(x)$ for the elements covering $x$ and…

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…

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…

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 $[m]\times[n]$ 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 $a$ 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…

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.

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…

Coxeter-element invariance conjecture. Any linear combination of the indicator functions $\chi_y$ for $y\in S$ is $c$-mesic or orbomesic under the action of $W$ if and only if it i…

Rank-alternating homomesy conjecture. The triple $(\mathcal{A}^{\ell}(\widetriangle_n),\operatorname*{TogPro},\mathcal{R})$ is $0$-mesic when $n$ is even and $\frac{\ell}{2}$-mesic…

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 $(P,Q)$ be a minuscule doppelgänger pair, and let $\mathcal{PP}^{m}(P)$ and $\mathcal{PP}^{m}(Q)$ denote their bounded plane-partition sets. Minuscule doppelgänger rowmotion co…

Fix positive integers $a,b,c$ and let $\mathsf{B}_{a,b,c}={\bf a}\times{\bf b}\times{\bf c}$ be the product of three chains. Let $J(\mathsf{B}_{a,b,c})$ denote its order ideals, an…

Down-degree homomesy conjecture. For every toggle on antichains-symmetric distribution $\mu$ on $J(\mathscr T(a,b))$,