433 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 $l$ 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 $(m_1,n_1)$ 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 $f_1,\ldots,f_m$ 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…

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 $\mathcal{A}$ 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…

Let ${\mathcal L}$ be a lattice and let $C$ be a crosscut of ${\mathcal L}$. Write $\Gamma(C,{\mathcal L})$ for the crosscut complex, $\operatorname{Bd}(\Gamma(C,{\mathcal L}))$ fo…

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…

Möbius support conjecture. For any base polymatroid $\mathscr{P}$, the Möbius support $\mu\text{-supp}(\mathscr{P})$ is a generalized polymatroid. The theorem preceding this conjec…

Let $n\in\mathbb{P}$, and let $\mu(u,v)$ denote the Möbius function of the Bruhat-order interval $[u,v]$ in the poset of signed Wachs permutations $\operatorname{\mathcal{W}}(B_n)$…

Let $P$ be the poset shown in Figure, and let $G(P)$ be its associated graph. Consider the coefficient-free $R$-system associated with $G(P)$. A strong $\tau$-sequence and the sing…

Saks-West conjecture. In every product $P\times Q$ of two posets, the size of a largest semiantichain equals the size of a smallest unichain covering.

Polynomial-time dimension conjecture. The dimension of a poset given its linear extension graph can be determined in polynomial time.

Let $P$subseteq be a finite poset. Let $\operatorname{Mon}(\mathbb{Z})$ be the set of functions $f\colon\mathbb{Z}\to\{0,1\}$ such that $f(n)=1$ and $f(-n)=0$ for all sufficiently…

Finite-generation conjecture. For every positive integer $n$, this closed cone is finitely generated.

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegativ…

Cone equality conjecture. The two cones are equal:

PL sphere-or-ball conjecture. If $[u,v]$ is full, then $(u,v)$ is a PL sphere of dimension $\rho(v)-\rho(u)-2$. Otherwise, $(u,v)$ is a PL ball of the same dimension.

Let $PC_n$ be the poset of unordered pairs $\{\alpha,\beta\}$ of compositions satisfying $|\alpha|+|\beta|=n$, ordered by … For an unordered pair $\{\alpha,\beta\}$, define…

Let $\mathcal{HA}_n$ be the poset of hyperarbres, and let $\operatorname{WH}_i(\mathcal{HA}_n)$ denote its Whitney homology groups. Define the symmetric-function generating series…