43 problems

Three-halves-unit shifted-hole tiling conjecture. The number of lozenge tilings of this hexagon with the triangular hole equals

Consider a lozenge tiling of a hexagon with normalized side lengths $\alpha,\beta,\gamma$, and define the arctic region to be the set of lozenges connected to the boundary by seque…

Let $\alpha,\beta,\gamma$ be the normalized side lengths of a hexagon, let $(x,y)$ be a normalized location in this hexagon, and let $\mathcal P_{\alpha,\beta,\gamma}(x,y)$ denote…

Twisted domino correspondence conjecture. For non-negative integers $m$ and $n\geq1$, there should be a bijection

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 $\CSPP{n}^{k}$ be the restricted class of cyclically symmetric plane partitions, let $\overline U_r(c)$ be its statistic for $1\leq r\leq n$, and define … where $\MT{n}^{k}$ is…

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…

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…

Let $G$ and $G'$ be the symmetry groups specified in the claim, and write $M_G(a,b,c)_{-1}$ and $A_G(a,b,c)_{-1}$ for the corresponding matrices specialized at $q=-1$. Let…

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 … , with arches connecting consecutive points, is $A_V(r)$. The sum-of-entries formula is established in the supplied discussio…

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and let $S_{2n-1}^{(1)}$ be the sum of the components of its groundstate wavefunc…

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and normalise its groundstate wavefunction so that its smallest element is $1$. L…

Consider the dense O(1) loop model with open boundary conditions and system size $L$, and let $S_L^{(1)}$ be the sum of the components of its groundstate wavefunction, normalised s…

Consider the dense O(1) loop model with open boundary conditions and system size $L$, and normalise its groundstate wavefunction so that its smallest element is $1$. Let $N_8(2n)$…

The quantum-invariant-chain 1-sum conjecture.

Let $G_{m,n}$ be the degree-$m$ part of the grove algebra, and let $c$ generate the cyclic action … The basis elements are to be indexed by height $m$ staircase plane partitions of…

Enumeration conjecture. The number of nondegenerate $2\times (k+1)\times k$ hypermatrices over $\mathbb{F}_q$ which respect $P$ is

Let $pl(n)$ denote the plane partition function, counting plane partitions of $n$. A perfect power is an integer of the form $a^b$ with integers $a,b>1$. The plane-partition perfec…

Let $\sigma_2(n):=\sum_{d\mid n}d^2$, and let $\Delta_n^{\sigma_2}(x)$ denote the difference associated with the D'Arcais polynomials for $g(n)=\sigma_2(n)$. Plane-partition positi…

Let $P_{\lambda/\mu/d}$ be the cell poset associated with a cylindric skew shape $\lambda/\mu$ of period parameter $d$, and let $\Omega(P_{\lambda/\mu/d};t)$ be its order polynomia…

For integers $n\ge 0$ and $\delta\in\frac12\mathbb Z$, let … where $\mathrm{PP}$ is the set of plane partitions and $w_-$, $w_0$, and $w_+$ are the sums of the entries strictly bel…

Inversion-enumeration conjecture.