73 problems

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…

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

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 on $n$ elements and write $\mathrm{e}(P)$ for its number of linear extensions. For $x,y\in P$, let $P+xy$ denote the poset obtained by adjoining $x<y$ and…

Let $P$ be a poset on $n$ elements, let $L(P)$ denote its set of linear extensions, and let the number of antichains of $P$ be at most $m$. Bucket orders are posets obtained by par…

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 with $n$ elements, let $x\in P$, and let $P-x$ be obtained by deleting $x$. Write $h_P(y)$ for the expected position of $y$ in a uniformly random linear e…

Let $P$ be a finite poset, let $D$ be an ideal, and put $U=P\setminus D$. Let $h(x)$ be the expected position of $x$ in a uniformly random linear extension, and let…

Let $P$ be a finite poset, let $w(P)$ denote its width, and let $\operatorname{gap}(P)$ be the maximum gap between consecutive expected positions $h(x)$, including the initial and…

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 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…

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…

Consider a finite poset $P=(\\{p_1,\dots,p_n\\},\leq_P)$. A linearization of $P$ is a linear order on $P$ compatible with $\leq_P$; let $L(P)$ be the set of linearizations and let…

Let $R\subseteq\mathcal{R}$ be a set of comparison results, and let $\mathcal{X}(R)$ be the set of permutations consistent with $R$. For items $j_1,j_2\in[m]$, consider the probabi…

A finite poset has height two if every chain has at most two elements. Height-two linear-extension conjecture. All but finitely many positive integers are the numbers of linear ext…

For a nonnegative rational number $\alpha$, let $\operatorname{s}(\alpha)$ be the sum of the quotients in its unique simple continued fraction. Kravitz–Sah continued-fraction conje…

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 $P=(X,\prec)$ be a finite poset that is not a chain, and let $f$ be uniformly random in the set $\operatorname{\mathcal E}(P)$ of linear extensions. One-third–two-thirds conjec…

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…

For each positive integer $n$, let … Width-two range conjecture. There exists a constant $c>1$ such that the upper end of the known interval in $\mathcal T_e(n)$ can be improved fr…