109 problems

Cyclic sieving conjecture for multi-triangulations. The polynomial $\Omega_{\delta_n}(m;q)$ is a cyclic sieving polynomial for the action of rotation on multi-triangulations. This…

Let $d_k(n)$ be defined by the generating function … In particular, $d_2(n)$ counts the relevant 2-elongated plane partitions. Andrews–Paule conjecture. For all integers $n\geq 0$…

Bender–Knuth conjecture. The generating function with respect to the norm of these strict plane partitions is

Quantum foam conjecture. The statistical model of random plane partitions is nothing but quantum Kähler gravity.

Let $\lambda/\mu$ be a skew shape with $\lambda_1\geq\ell(\lambda)$, and let $P_{\lambda/\mu}$ be its cell poset. For $S=\{0,\ldots,\lambda_1\}$, write $M[S]$ for the class of poly…

Proctor's conjecture. The minuscule posets are the only posets for which $F_P(m;q)$ has a product formula of this form.

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

Totally symmetric plane-partition determinant conjecture. For every nonnegative integer $n$,

A plane partition is a filling of a finite box by cubes, and a symmetry class is the set of plane partitions invariant under one of the symmetries of the box, possibly including co…

Let $[m]$ denote the list $1,\dotsc,m$, and let $F_{[m],[n]}(x)$ be the polynomial defined through the factorisation and recurrence relations for the tiling generating functions of…

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…

Let $n$ be a positive integer, and consider a semiregular hexagon with side lengths $2n-1$, $2n-1$, and $2n$, repeated in opposite positions. A central lozenge is the lozenge occup…

A plane partition is a finite set $\mathcal{P}\subseteq \mathbb{N}^3$ of lattice points with positive integer coordinates such that if $(r,s,t)\in\mathcal{P}$ and $1\leq i\leq r$,…

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

Let $\CSPP{n,2}$ denote the relevant generalized class of cyclically symmetric plane partitions, and let $|c|$ be the size of $c$. For $k\geq0$, define the product appearing below.…

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…

Let $\{a_j\}_{j\geq 1}$ be a sequence of integers, and let … be its Dirichlet series, with parameter $\alpha$ determined by convergence in the half-plane…

Limit surface conjecture. For each continual Young diagram $f$, there should exist a limit surface $L_f$ defined on $D_f$, bounded between the $x$-axis and the graph of $f$, that d…

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…

Enumeration conjecture. The weighted enumeration equals, up to sign, the following quantity, according to the congruence classes of $a$, $b$, and $c$ modulo $4$:

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 $H_n(x,y)$ 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-…