63 problems

At the Razumov–Stroganov point $q=e^{2i\pi/3}$, take the homogeneous limit $w_i=1$ for all $i$ in the level-one $D_r$ $q$KZ solution, viewed as the Perron–Frobenius eigenvector…

At the Razumov–Stroganov point $q=e^{2i\pi/3}$ and in the homogeneous limit $w_i=1$, let $\Psi$ be normalized so that its smallest entry is $\Psi_{\pi_0}=1$. Let $v$ be the left ei…

Determinant-evaluation conjecture. The following identities should hold:

Mills–Robbins–Rumsey's refined vertical-symmetry conjecture. The number of $b\in\TSPP{2n+1}$ satisfying $\gamma(b)=b$ and $U_2(b)=r-1$ should equal the number of vertically symmetr…

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…

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…

Vertically symmetric nest distribution conjecture. The nest distribution is

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…

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…

At $q=e^{2i\pi/3}$ and in the homogeneous limit, let … for $C_r$ is the sum of entries for $C_{r-1}$. The assertion is presented after empirical sum and pairing formulas, and the s…

At $q=e^{2i\pi/3}$, let … is … For odd $r=2n-1$, the preceding discussion presents the corresponding product as computed, while the even case follows by a specialization from the o…

At $q=e^{2i\pi/3}$ and in the homogeneous limit, let … , with arches connecting consecutive points, is $A_V(r)$. The sum-of-entries formula is established in the supplied discussio…

Let … , corresponding to the link pattern with arches connecting consecutive points, is $A(n-1)$. This refines the known result that the sum of entries is $A(n)$; the largest-entry…

For the rotor model with closed boundary conditions, let $\psi_0$ be its groundstate wavefunction and let $S(N)$ denote the sum of its elements. For odd size $N=2m-1$, normalize th…

Vertically symmetric ground-state conjecture. The vector

Half-turn-symmetric ground-state conjecture. The vector

Let $|\Psi^{(2n)}\rangle$ be the ground-state vector for the twisted XXZ chain with $2n$ sites, let $W^{(2n)}$ index its components, and let $A_n$ be the number of alternating-sign…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, let … and let $A_n$ be the number of alternating-sign $n\times n$ matrices…

Let $|\Psi^{(2n)}\rangle$ be the normalized ground-state vector of the twisted XXZ chain with $2n$ sites, and let $A_n$ denote the number of alternating-sign $n\times n$ matrices.…