Htydjtk

Partially ordered sets (posets) are important objects in combinatorics (with basic connections to extremal combinatorics and to algebraic combinatorics) and also in other areas of mathematics. They are also related to sorting and to other questions in the theory of computing. I am asking for a list of open questions and conjectures about posets.

The 1/3-2/3 conjecture is probably considered one of the most significant open problems about finite posets; see the Wikipedia page: https://en.wikipedia.org/wiki/1/3%E2%80%932/3_conjecture.

Here's a problem that I believe has little chance of being resolved, and it's also not so clear to me what the motivation behind the problem is, but it involves some very pretty algebraic combinatorics.

Stanley (in his PhD thesis) defined a poset $P$ to be Gaussian if for every $mgeq 0$ we have
$$ sum_{I in J(Ptimes [m])} q^{#I} = frac{prod_{i=1}^{r}(1-q^{h_i+m})}{prod_{i=1}^{r}(1-q^{h_i})}$$
where $r$ and $h_1,ldots,h_r$ are constants independent of $m$. (Here $J(P times [m])$ is the distributive lattice of order ideals of the Cartesian product of $P$ and the chain poset $[m]={1,2,ldots,m}$.)

Using results from Standard Monomial Theory, in "Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations", Proctor proved that if $P$ is a minuscule poset, then $P$ is Gaussian. Here a minuscule poset is a certain finite poset coming from a minuscule representation of a semisimple Lie algebra; these have been classified- see Proctor's paper for the list. Furthermore, Proctor conjectured that the minuscule posets are the only Gaussian posets.

I think a few things are known (due to some mixture of Proctor and Stanley) about this conjecture: that a Gaussian poset $P$ is a disjoint union of connected Gaussian posets; that a connected Gaussian poset has to be graded; that $r$ has to be the number of elements of $P$; that the $h_i$ have to be the ranks of the elements $pin P$; maybe some more things too. But like I said I think the full conjecture is pretty hopeless.

Here's another classic from algebraic combinatorics.

In his PhD thesis "Ordered Structures and Partitions", Stanley introduces the $(P,omega)$-partition generating function of a labeled poset $(P,omega)$. This is defined to be $K(P,omega) := sum_{sigma in A^r(P,omega)} x^{sigma}$, where $A^r(P,omega)$ is the set of all reverse $(P,omega)$-partitions (i.e., fillings of the poset $P$ with entries in $mathbb{N}$ obeying certain inequalities depending on $omega$ and the order structure of $P$), and where we use the notation $x^f := prod_{i geq 1} x_i^{#f^{-1}(i)}$. Stanley shows that $K(P,omega)$ is always a quasisymmetric function. When $P$ is the poset of a skew Young diagram and $omega$ is a compatible ``Schur labelling'' then $K_{(P,omega)}$ is in fact a symmetric function (namely, a skew Schur funciton). Stanley's conjecture is that $K_{(P,omega)}$ is a symmetric function if and only if $(P,omega)$ is isomorphic to $(P_{lambda/mu},w)$, where $lambda/mu$ is a skew-shape and $w$ is a Schur labelling of $P_{lambda/mu}$. There has been some progress on this conjecture due to Malvenuto; I don't know the exact status of what has been proved.