434 problems

Let $\beta(L)$ denote the beta-value associated with a list $L$, and let list concatenation be written by juxtaposition. Let $m,n$ be non-negative integers with $n>m$, and let $L$…

Let $(m_1,n_1)$ and $(m_2,n_2)$ be pairs of non-negative integers, where $(m_1,n_1)$ is strictly better balanced than $(m_2,n_2)$, meaning that the first pair is strictly closer to…

Let $\mathcal{T}_n$ be the set of all trees with $n$ vertices, ordered by $\succeq$. The length of a chain is the number of elements in a sequence $T_1\succ T_2\succ\cdots$ under t…

Let $([d],\mathcal{L})$ be a covering clutter, let $s$ be its magic subspace, and let a magic permutation be a permutation of $[d]$ realizable by a positive point in $s$. For a per…

A hull configuration is a point configuration lying either on a line segment or on the boundary of a convex polygon; a blank side is an edge of the convex hull containing only two…

Kahn's conjecture. If $|X|=|Y|=|Z|$, then

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for linear extensions of the rectangular poset with parameters $s,t$ that avoid the permutation pattern $\sigma$,…

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for pattern-avoiding linear extensions of the rectangular poset with parameters $s,t$. The 2143 two-column invers…

Let $l$ be a positive integer, and let $i$ and $j$ be non-negative integers with $j>i$. For $1\leq k\leq l$, let $L_{i,j}^k$ be the list of length $l$ whose entries are $i$ except…

Let $f_1,\ldots,f_m$ be positive linear forms on $\mathbb{R}^d$, and let $s$ be the subspace on which they all vanish. For a permutation $\sigma$ of $[d]$, define the reverse domin…

The supplied sources identify the Gold Partition Conjecture as a conjecture about finite posets and describe it as a strengthening of the $1/3$–$2/3$ conjecture, but they do not pr…

Let $\mathcal{A}$ be a finite simplicial hyperplane arrangement of rank $r$, and let $W(\mathcal{A})$ be its poset of regions with the standard region order. Is…

Kapranov–Voevodsky conjecture. The map $f$ is a quotient by a weak order congruence. This conjecture asks whether the higher-dimensional analogue of the map from weak Bruhat order…

Brenti's conjecture. For any labeled poset $(P,\omega)$, the polynomial $W(P,\omega;t)$ has only real zeros as a polynomial in $t$. This is the Poset Conjecture, also called the Ne…

Let $P$ be a finite poset. Write … and … Define $\pi(P)=\lim_{n\to\infty}La(n,P)/\binom{n}{\lfloor n/2\rfloor}$ and…

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

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, 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…

Let $d6f2$ be a Coxeter diagram of type $D_n$ with $n\geq 3$. Construct the $(s_n,s_{n-1})$-subdivision $d6f2'_d6f2$ of the simplicial complex $d6f2_d6f2$ by subdividing each edge…

For a permutation $w$, let $\rp(w)$ denote its set of reduced pipe dreams, and order it by generalized chute moves. Rubey's lattice conjecture. This poset is a lattice. Certain spe…

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

Brenti's conjecture. The polynomial $W_{P,\omega}$ is log-concave with no internal zeroes.

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

Foley–Hoàng–Merkel's twinning conjecture. If $P$ is $(3+1)$-free and $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive, then $X_{\operatorname{Inc}(P_v)}(\mathbf{x})$ is $e$-…

Quasi-reductivity conjecture. An index-one Lie poset algebra of type A, B, C, or D is contact if and only if it is quasi-reductive.