36 problems

Let $[m]^n$ denote the $n$-dimensional grid poset, and let $\bar{s}([m]^n)$ be its average jump number. The parameters $m$ and $n$ range over values for which…

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\{y:y\not\sim x\}$ and define the range parameter … For distinct $x,y\in P$, let…

Let $P$ be a finite poset, let $w(P)$ denote its width, and for distinct elements $x,y\in P$ define … with … where the probabilities are over a uniformly random linear extension of…

Let $P$ be a finite poset. For distinct elements $x,y\in P$, define … and let … where the probabilities are taken over a uniformly random linear extension of $P$. The 1/3-2/3 conje…

Let $P$ be a finite poset with $n$ elements, let $x,y\in P$ with $y\ne x$, and let $h_P(y)$ denote the expected position of $y$ in a uniformly random linear extension of $P$. The l…

Let $P$ be a finite poset, let $A$ be an ideal of $P$, and let $h(x)$ denote the expected position of $x$ in a uniformly random linear extension. Write $\max(A)$ for the maximal el…

Let $P$ be a finite poset, let $h(x)=\mathbb E f(x)$ for a uniformly random linear extension $f$ of $P$, and let $\operatorname{gap}(P)$ be the maximum consecutive gap among the or…

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\\{y\in P\setminus\\{x\\}:y\not\sim x\\}$, let $\pi(x)=|\Pi(x)|$, and let $\pi(P)=\max_{x\in P}\pi(x)$. Let $\delta(P)$ denote…

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 $P$ be a finite poset that is not a chain. For distinct $x,y\in P$, write $\delta_{xy}=\min\\{p(x\prec y),p(y\prec x)\\}$ and $\delta(P)=\max_{x\ne y}\delta_{xy}$. The 1/3–2/3…

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

Let $P$ be a finite poset that is not a chain. Consider two consecutive comparisons, and let $t_0=e(P)$, where $e(P)$ is the number of linearizations of $P$; let $t_1$ be the numbe…

Let $\mu(n)$ denote the minimum number of elements in a finite poset with exactly $n$ linear extensions. Kravitz–Sah conjecture. … This would improve the known bound…

Let $m$ be a positive integer. Height-two universality conjecture. For all sufficiently large $m$, there exists a poset $P$ of height two such that $e(P)=m$. The source states that…

Let $P$ be a finite sign-balanced poset, and let $\mathcal G(P)$ be the graph whose vertices are the linear extensions of $P$ and whose edges correspond to adjacent switches. Ruske…

Let $P$ be a finite poset with fixed prescribed values represented by $\mathbf z,\mathbf c$, and let $N_{\mathbf z,\mathbf c}(P)$ denote the number of linear extensions satisfying…

Let $e(P)$ be the number of linear extensions of a finite poset $P$, and let $e'$ denote its restriction to posets of height two. Write $\mathcal T_{e'}$ for the set of values atta…

A d-complete poset is a poset in the sense of the standard definition of d-completeness. A poset is LE-cactus when the Bender–Knuth involutions acting on its linear extensions sati…

Let $F_\lambda$ be the Ferrers poset associated with a partition $\lambda$, and let $\lambda^t$ be the conjugate partition; the posets $F_\lambda$ and $F_{\lambda^t}$ are isomorphi…

For even $n$, let $Z_n$ be the poset on $n$ elements with relations … A poset is LE-symmetric when its Bender–Knuth group acts as the full symmetric group on its set of linear exte…

Let $P$ be a d-complete poset. A poset is LE-cactus when its Bender–Knuth group action on its linear extensions satisfies the cactus relations. The d-complete-poset LE-cactus conje…

Let $P=(X,\prec)$ be a poset with $|X|=n$ elements. Fix a nonempty subset $A\subseteq X$, and let $f$ be chosen uniformly from the set of linear extensions of $P$. Write…

Let $\mathcal G=(V\sqcup W,E)$ be the bipartite graph defined above, with the parts and edges constructed from the relevant sets of linear extensions. Matching conjecture. There ex…

Let $P=(X,\prec)$ be a finite poset with $n=|X|$, and let $x,y\in X$ be distinct. Denote by $\operatorname{F}(k)$ the number of linear extensions…

Assume the conditions of the generalized cross-product conjecture, so that $F(i,j)$ counts the relevant linear extensions for integer indices. Generalized cross-product inequality.…