125 problems

An alternating sign matrix (ASM) of order $n$ is an $n\times n$ matrix whose entries are in $\{0,1,-1\}$ and whose nonzero entries in each row and column alternate, beginning and e…

Let $n\geqslant 1$ be an integer, and consider the ground-state eigenvector of the periodic XXZ spin chain with anisotropy parameter $\Delta=-\frac12$ and $N=2n+1$ sites. Normalize…

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…

Di Francesco's conjecture. The number of domino tilings of the Aztec triangle is given by a product formula similar to the product formula for alternating sign matrices.

Vertically symmetric nest distribution conjecture. The nest distribution is

Let $L$ be an even circumference, let $P[c]$ denote the probability of connectivity state $c$, and let $A_{\mathrm{HT}}(L)$ denote the number of $L\times L$ alternating sign matric…

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

Determinant-evaluation conjecture. The following identities should hold:

Let $\TSPP{n}^{k}$ be the subset of triangular shifted plane partitions whose entries in the first $n-1-k$ columns attain their maximal values $n$. Let $\MT{n}^{k}$ be the set of m…

An alternating sign matrix is a square matrix whose entries lie in $\{-1,0,1\}$, with each row and column summing to $1$ and with the nonzero entries alternating in sign. Let $A_n$…

Enumeration conjectures for odd-order HTSASMs and DSASMs. The number of $(2n+1)\times(2n+1)$ HTSASMs 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…

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…

Let $P_n(x)$ be the generating function for alternating sign matrices invariant under flips in the vertical and horizontal axes, with the weight defined in the source. Such matrice…

Consider self-complementary cyclically symmetric plane partitions whose Ferrers graph lies in ${\cal X}_{2n}=[1,2n]\times[1,2n]\times[1,2n]$, with a part $a_{ij}$ called special wh…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric alternating sign matrices. Quarter-turn polynomial factorization conjecture. There exists a sequence of polynom…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric $n$ by $n$ alternating sign matrices, and let $H_n$ and $A_n$ denote the half-turn-symmetric and unrestricted a…

Let $H_n(x,y)$ be the generating function for half-turn-symmetric alternating sign matrices, and let $R_n(x,\mu)$, $T_n(x,\mu)$, and the polynomials $S_{4n+1}(x),S_{4n-1}(x)$ be as…

Let $H_n(x,y)$ be the weight-generating function for half-turn-invariant $n$ by $n$ alternating sign matrices, and let $Z_n(x,y,0)$ and $Z_{n-1}(x,y,1)$ be the corresponding plane-…

Let $F_n(x)$ be the weight-generating function for flip-symmetric alternating sign matrices, with the exponent recording the number of $-1$ entries in the first half of the columns…

Let $A_n(x,y)$ be the weight-generating function for $n$ by $n$ alternating sign matrices, where a matrix with $r$ entries equal to $-1$ and its top-row $1$ in position $s$ has wei…

Let an alternating sign matrix of order $n$ be represented by its colored square-ice graph, and let the $2n$ blue endpoints be numbered cyclically. The pairing of an alternating si…

For an odd lattice size $2n+1$, let $\Phi_{2n+1}^{(s)}$ be the maximally nested component with red arcs joining $(i,2n+1-i)$, green arcs joining $(i+1,2n+2-i)$, a red line emanatin…

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…

Let $\Psi^{(L)}(\tau)$ be the homogeneous open boundary qKZ solution, and let $\pi_{\max}$ connect points $2i-1$ to $2i$, with the last point unmatched when $L$ is odd. Let…