26 problems

Zuber's conjecture. The number $A_{X,Y}(m)$ is

Zuber's conjecture. The number $A_X(m)$ is

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…

Vertically symmetric nest distribution conjecture. The nest distribution is

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

Geometric FPL expansion conjecture. The pushforward can be decomposed as

Geometric Razumov–Stroganov conjecture. The pushforward of $\sigma_r$ to $\operatorname{Mat}_<(N)$ in $T$-equivariant $K$-theory is equal, up to normalization, to

Let $i,j,\ell,m\geq 0$, and let $A_X$ denote the number associated with a link pattern $X$ in the paper's chord-diagram notation. Wieland-type chord-diagram conjecture. The equalit…

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…

Negative-root multiplicity conjecture. All real roots of $\psi_\pi(t)$ are negative integers, and $-p$ occurs with multiplicity $m_p(\pi)$. Equivalently,

Let $\pi$ be a matching, let $d(\pi)$ be the degree parameter used in the source, and let $P_\pi(\tau,t)$ be the associated bivariate polynomial. Tau-deformed coefficient-positivit…

For a matching $\pi$, define $G_\pi(\tau)=\Psi_\pi(\tau,-|\pi|)$. Let $d(\pi)$ be the degree parameter used in the source, and let $\Psi_\pi(-\tau)$ denote the specialization…

Let $\pi$ be a matching and let $p$ be an integer with $1\leq p\leq|\pi|-1$ such that $m_p(\pi)=0$. Write $\pi=\alpha\circ\beta$ with $|\alpha|=p$, and define…

Let $\pi$ be a matching, and consider the bivariate polynomial $\Psi_\pi(\tau,t)$ as a polynomial in $t$ with coefficients in $\mathbb{Q}[\tau]$. Let $d(\pi)$ and $m_i(\pi)$ have t…

Let $\pi$ be a matching and let $A_\pi(t)$ be its polynomial. Coefficient-positivity conjecture. Every coefficient of $A_\pi(t)$ is nonnegative. The conjecture is supported by comp…

For a matching $\pi$, define $G_\pi=A_\pi(-|\pi|)$; let $A_n$ and $A_n^V$ denote the enumerative quantities used in the source, and let $^n$ denote the empty matching with $n$ nest…

Let $\pi$ be a matching and let $p$ be an integer with $1\leq p\leq|\pi|-1$ such that $m_p(\pi)=0$. Write $\pi=\alpha\circ\beta$, where $|\alpha|=p$, and define…

Let $\pi$ be a matching, let $A_\pi(t)$ be its polynomial, let $d(\pi)$ denote the relevant degree, and let $m_p(\pi)$ denote the multiplicity assigned to the integer $p$. Real-roo…

Razumov–Stroganov conjecture. For every link pattern $pi$ of size $2n$,

For half-turn symmetric FPL diagrams with $L=2n$ external terminals, let $P(L,m)$ count diagrams with $m+1$ nests and define … Let $Z_{\rm HTSFPL}(2n)$ be the total number of half-…

Let an FPL diagram of size $L$ have size $(L-1)\times L/2$ when $L$ is even and size $L\times (L-1)/2$ when $L$ is odd. Let $d$ be its depth, and let $S(L,d)$ denote the total numb…

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…

For each size $n$, let $A(n;y)$, $A_{\textsc{HT}}(n;y)$, and $A_{\textsc{QT}}(n;y)$ denote the enumerating polynomials of FPLs, HTFPLs, and qQTFPLs, respectively. Give an object we…