21 problems

For integers $m\geq 1$ and $1\leq r\leq m-1$, let $d53c^{(m,r)}_n$ be the set of greedy $(m,r)$-Tamari intervals of size $n$, and let … be their generating function with the weight…

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 $L_n^d$ be the lattice of $d$-torsion classes associated with the $(P,\phi)$-Tamari lattice, and let $J(os_n^{d+1})$ denote the set of order ideals of the poset $os_n^{d+1}$. F…

The $m$-Tamari lattice and the linear type $A$ $m$-Cambrian lattice are lattices associated with the same parameter $m$. Stump's interval-count conjecture. The $m$-Tamari lattice a…

Let $\lambda$ be a triangular partition, and let its top-down tableau be the Young tableau associated with a maximal chain in the $\nu$-Tamari lattice. Say that this tableau is sim…

Maximal-degree subposet conjecture.

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

Cover enumerator conjecture.

For integers $m,t\geq 1$, let $h_t(m,n)$ denote the number of $t$-$\operatorname{\mathsf{Pop}}$-sortable $m$-ballot paths in the $m$-Tamari lattice $\operatorname{Tam}_n(m)$. Ratio…

F-triangle conjecture. The rational function $F_{\alpha}(x,y)$ is a polynomial with nonnegative integer coefficients if and only if $\alpha$ has at most one part exceeding $1$.

Let $n>0$ and let $\alpha$ be a composition of $n$. Let $\mathcal{T}_{\alpha}$ denote the parabolic Tamari lattice, and let $\operatorname{CLO}(\mathcal{T}_{\alpha})$ denote its co…

Let $n>0$ and let $\alpha$ be a composition of $n$. Let $\mathcal{T}_{\alpha}$ denote the parabolic Tamari lattice, and let $\operatorname{CLO}(\mathcal{T}_{\alpha})$ denote its co…

Let $n>0$ and let $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ be a composition of $n$ into $r$ parts. Let $H_{\alpha}(x,y)$ be the polynomial defined on order ideals of the releva…

Let $\operatorname{\mathbf{Tam}}_n$ be the Tamari lattice, and consider its intervals. For an interval, measure degree separately in the variable pairs $(x,y)$ and…

Let $\operatorname{\mathbf{Tam}}_n$ be the Tamari lattice, and let $\mathbb{D}_n$ be its refined interval polynomial in variables $x,y,\overline{x},\overline{y}$. An interval has d…

Enumeration conjecture. For all $i\geq-1$,

Let $\mathcal{T}_{k,n}$ denote the Graßmann-Tamari poset on the relevant Graßmannian tableaux. The usual Tamari order is recovered when $k=2$. Graßmann-Tamari lattice conjecture. F…