8 problems

Let $\boldsymbol{z}$ be generic on the unit circle, let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function, and write $\#\operatorname{LP}(n,m)$ for the number of…

Let $\mathcal{QD}(\boldsymbol{z})$ be the specified space of quadratic differentials with $m$ poles and $n$ zeros, and let $\Gamma(Q)$ denote the horizontal trajectories of…

A fully packed loop configuration (FPL) has occupied external edges whose connections form a link pattern. Let $\pi$ be a fixed link pattern of size $n$, and let $\pi\cup m$ denote…

Let $\mathcal{D}_n$ be the set of link patterns of size $n$, and let $c_{\alpha\pi}$ be the coefficients introduced in the paper. Write link patterns as binary words, with…

Let $\mathcal{D}_n$ be the set of link patterns of size $n$, let $\mathbf{1}_n$ denote the maximal link pattern, and let $c_{\alpha\pi}$ be the coefficients introduced in the paper…

Let $A_{\textsc{QT}}(4n+2;w)$ denote the number of qQTFPLs of size $4n+2$ with the induced half-turn-invariant link pattern $w$ of length $2n+1$, and let $A_{\textsc{HT}}(2n+1;w)$…

Let $A_{\textsc{QT}}(4n;w)$ denote the number of quarter-turn-invariant FPLs of size $4n$ with link pattern $w$, where $w$ is a bilateral Dyck word of length $2n$, and let…

Let $A_{\textsc{HT}}(N;\pi)$ be the number of half-turn-invariant FPLs of size $N$ with half-turn-invariant link pattern $\pi$, and let $A_{\textsc{HT}}(N)$ be the total number of…