707 problems

Let $P$ be a poset, and let $W_P(t)$ denote its $W$-polynomial, also called its $P$-Eulerian polynomial. Neggers–Stanley conjecture. The polynomial $W_P(t)$ is real rooted. This co…

Let $\rho$ be a permutation, and let $D(\rho)$ be the principal downset of all permutations contained in $\rho$, ordered by permutation-pattern containment and ranked by permutatio…

Let $P$ be a finite poset on $n$ elements, and let $\mathcal{O}(P)$ and $\mathcal{C}(P)$ denote its order polytope and chain polytope, respectively. For each $k=0,\dots,n-1$, write…

Tilted combinatorial invariance conjecture. Then

Let $w$ be a permutation, and let $\mathsf{PD}(w)$ be the set of pipe dreams for $w$ equipped with the chute-move partial order $\mathrel{\leq_{\mathrm{chute}}}$, where…

Lonc's truncated partition conjecture. If $n$ is sufficiently large and $|P|$ divides $2^n-2$, then $2^{[n]}-\{\emptyset,[n]\}$ has a $P$-partition.

For a finite poset $(P,\leqslant)$, let $e(P)$ be the largest integer such that the union of any $e(P)$ consecutive layers of $2^{[n]}$ is weak $P$-free, and let…

Let $P$ be a finite poset, let $w(P)$ denote its width, and let $\delta(P)$ be the maximum, over distinct $x,y\in P$, of $\min\\{p(x\prec y),p(y\prec x)\\}$. The Kahn–Saks conjectu…

Füredi's conjecture. For every positive integer $n$, the Boolean lattice $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that every chain has size eit…

A poset $P$ satisfies the finite antichain condition (FAC) if it has no infinite antichain. A partition into antichains is a partition of $P$ whose parts are antichains, and a chai…

Ferroni–Matherne–Vecchi conjecture. The Chow polynomial $H_P(t)$ is real-rooted.

Morier-Genoud–Ovsienko conjecture. The rank polynomial of $J(P)$ is unimodal.

Let $P$ be a poset on $n$ elements, and let $\Omega(P;t)$ be its order polynomial. Kahn–Saks conjecture. The map … is weakly decreasing for positive integer values of $t$. The conj…

Shellability conjecture. The intervals in the level Eulerian poset from Section are shellable, and hence their order complexes are homeomorphic to spheres.

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…

Neggers' conjecture. These $(P,\omega)$-Eulerian polynomials have only real roots.

Let $Q_n$ be the poset of all subsets of $\\{1,\ldots,n\\}$ ordered by inclusion. A chain decomposition partitions $Q_n$ into chains, and two decompositions are orthogonal if every…

Let $\Omega$ be a Gorenstein poset, and let its cd-index be the homogeneous noncommutative polynomial in $\mathbf{c}$ and $\mathbf{d}$ associated with its flag $f$-vector, where…

Weak-order lattice conjecture. The weak order of $(G,F)$ always has the structure of a lattice.

Brenti–Welker's conjecture. The chain polynomial of the face lattice of every polytope is real-rooted.

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…

Chan–Pak conjecture. For every $A\subseteq P$, the sequence of probabilities of the minimum position is log-concave:

Let $P$ be a poset with $n$ elements. A labeling of $P$ is tangled if it requires $n-1$ applications of extended promotion to become a natural labeling. Defant–Kravitz conjecture.…

Let $L(m,n)$ denote the finite Young lattice of partitions that fit inside an $m\times n$ rectangle. A symmetric chain order is a partition of the poset into chains symmetric about…

Twin-free reduction conjecture. If $P$ is $(3+1)$-free and twin-free, then $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive.