50 problems

Fully Packed Loops (FPL) are obtained by coloring edges of a square grid so that exactly two edges incident to each inner vertex are colored, while every other boundary edge is col…

Let $\pi_a$ range over link patterns on $2n$ external links, let $A_n(\pi_a)$ be the corresponding FPL numbers, and let … be the eigenvector of the periodic Temperley–Lieb Hamilton…

Vertically symmetric nest distribution conjecture. The nest distribution is

Refined Razumov–Stroganov conjecture. Every entry $\boldsymbol{\Psi}_n^{(1)}(\pi\mid t)$ is a polynomial in $t$ of degree at most $n-1$, and

A fully packed loop (FPL) configuration is a fully packed loop on a square grid, and its link pattern records the induced noncrossing matching of boundary points. Consider link pat…

Zuber's polynomiality conjecture. The quantities $A_{(\pi)_p}$ and $\psi_{(\pi)_p}$ are polynomials in $p$.

Write $p$ repeated opening parentheses as $(^p$ and $p$ repeated opening parentheses followed by $p$ closing parentheses as $(\ldots)_p=(^p\ldots)^p$. For matchings of the form…

Stationary-state enumeration conjecture. For any of these four triples, $a_F$ equals $M_F(G)$, the number of fully packed loop diagrams on $G$ with matching $F\in{\mathcal F}$. Thi…

Link-pattern balance conjecture. For any $n=1,2,\ldots$, one has

Let $\Psi_\pi(2n)$ be the components of the open-boundary qKZ solution indexed by link patterns $\pi$, and let $\operatorname{VSFPL}_{2n+1}(\pi)$ count vertically symmetric fully p…

Asymptotic conjecture. When $p$ is large with $r$ fixed, or $r$ is large with $p$ fixed, one has

Even two-boundary FPL conjecture. The coefficients of the ground state enumerate fully packed loop configurations with the appropriate link patterns on the $(L+1)\times L$ patch an…

Componentwise Razumov–Stroganov conjecture. Each coefficient of $\psi_0(1)$ is related to the number of fully packed loop configurations, or equivalently symmetric alternating-sign…

VHASMs ground-state sum conjecture. For $a=1$, the sum of the components of $\psi_0(1)$ for a system of size $L$ equals

Doubly refined two-boundary conjecture. The ground state $psi_0(a,b)$ is related to weighted fully packed loop diagrams on an $(L+1)\times L$ grid with special boundary conditions.

One-boundary ground-state conjecture. The ground state of a Markovian Hamiltonian with one boundary operator, acting in a link-pattern space, is related to vertically and horizonta…

Let $A_n$ be the total number of FPL configurations of size $n$, and let the two configurations shown in the source's Figure 3 define the corresponding inclusive sums. Additional i…

Let $A_n(\pi)$ denote a configuration number for a link pattern of size $n$, and let $C_n$ and $D_n$ denote the inclusive sums appearing in the source's Figure 2. Further inclusive…

Let $\pi$ be a link pattern described by two Young diagrams $(Y,Y')$, with $r$ and $r'$ arches respectively and $n-r-r'$ parallel arches between them. Let $|Y|$ and $|Y'|$ denote t…

Let $\pi$ be a link pattern described by a Young diagram $Y$, let $|Y|$ be the number of boxes of $Y$, and let $\operatorname{dim}Y$ denote the dimension of the corresponding irred…

Let $A_n(\pi)$ denote the number of fully packed loop configurations with link pattern $\pi$, and let $p,q,r$ be the parameters describing the link-pattern class shown in the sourc…

Three-parameter FPL enumeration conjecture. For $p,q,r\geq0$, the corresponding configuration number is given by the displayed factorial-ratio formula in the source, equivalently b…

Periodic FPL conjecture. The state $\ket0$ is obtained by counting fully packed loop configurations on an $L\times L/2$ rectangular grid. The conjecture was checked in the source t…

The $O(n_1,n_2)$ model is the version of the fully packed loop model in which the type $b$ vertex is forbidden. Its height description uses two independent height directions, corre…

Geometric FPL expansion conjecture. The pushforward can be decomposed as