32 problems

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 $P=[m]\times[n]$ 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…

Let $F$ be a fence with $t$ segments. Let $I_H(F)$ be its ideal-statistic homomesic space and $I_T(F)$ its ideal-toggleability space. Full-range codimension conjecture. For every i…

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…

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…

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

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…

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 $a\leq b$, and let $T(a,b)$ be the trapezoid poset … Under rowmotion on $T(a,b)$, consider the antichain cardinality statistic. Hopkins's conjecture. The poset $T(a,b)$ should…

Let $\Phi$ be a crystallographic root system with positive-root poset $\Phi^+$, whose order ideals are denoted by $\mathcal{J}(\Phi^+)$. An antichain is a subset of $\Phi^+$ whose…

Let $\Phi^+(A_n)$ be the positive-root poset of type $A_n$, and let the rank-alternating order ideal cardinality statistic be … for an order ideal $I$ of $\Phi^+(A_n)$. Bernstein–S…

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…

Constant-parameter homomesy conjecture.

Birational type B homomesy conjecture. The statistics $h^{\mathrm{B}}_i\colon \mathcal{A}^{\mathrm{B}}_{\kappa}(\mathsf{B}^n)\to \mathbb{R}_{>0}$, for $1\leq i\leq 2n-1$, and…

The type B piecewise-linear homomesy conjecture. The statistic $h^{\mathrm{PL}}_*\colon \mathcal{A}^{\mathrm{PL}}_{\kappa}(\mathsf{B}^n)\to \mathbb{R}$ is $\kappa$-mesic for…

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 $\mathfrak{S}_n$ be the symmetric group, and let $\tau$ denote the reversal-rotation Foatic action on $\mathfrak{S}_n$. For a permutation $w$, let $\operatorname{Fix}(w)$ be th…

Let $\mathfrak{S}_n$ be the symmetric group, and let $\overline{\rho}$ denote the complement-rotation Foatic action on $\mathfrak{S}_n$. For $w\in\mathfrak{S}_n$, let…

Let $\mathfrak{S}_n$ be the symmetric group, and let $\overline{\gamma}$ denote the complement-inversion Foatic action on $\mathfrak{S}_n$. For a permutation $w$, let…

Let $D_i(w)$ be the indicator that $i$ is a descent of $w\in\mathfrak{S}_n$. Consider the Foatic intertwinings discussed in Sheridan Rossi's dissertation. Sheridan Rossi's conjectu…

Let $w\in\mathfrak{S}_n$. Define the weak-excedance statistic $\operatorname{wexc}(w)$ to be the number of indices $i\in[n]$ such that $w_i\geq i$. The complement-rotatedInverse an…