489 problems

Let $\Sigma_0$ and $\Sigma_1$ be homotopy spheres, and let $\Sigma_0\subset T^*\Sigma_1$ be a Lagrangian embedding. A bundle of rigid tubes over $\Sigma_1$ is a tube bundle of the…

For integers $m\geq 1$ and $1\leq r\leq m-1$, let $d53c^{(m,r)}_n$ be the set of greedy $(m,r)$-Tamari intervals of size $n$, and let … be their generating function with the weight…

Let $\rho$ be a finitely based permutation class, and let $f_\rho(x)$ be its generating function. Noonan–Zeilberger conjecture. Every finitely based permutation class has a D-finit…

Let $N(w,d)$ denote the number of primitive multiple zeta values (MZVs) of weight $w$ and depth $d$, and let $x$ and $y$ be formal variables. Broadhurst–Kreimer conjecture. The gen…

Let $G(x,y,t)$ be the full generating function for Gessel walks with step set $\\{\nearrow,\swarrow,\leftarrow,\rightarrow\\}$, and let $G(0,0,t)$ be the generating function of Ges…

Andrews–Bachraoui conjecture. There holds

Guo–Miller rational strategy conjecture. Every lattice game has a rational strategy.

Minimal-period conjecture. The minimum period of $c_j(k)$ is $p_j$ for every $0\leq j\leq d$. This conjecture predicts that the divisibility data of the $a_i$ determine the exact p…

Symmetry conjecture. For all $m \ge 1$ and all $l_1, \ldots, l_m \ge 1$, the generating functions for the even involutions in $I(3412, [l_1, \ldots, l_m])$ and for the odd involuti…

Let a finite group model be a lattice-walk model with finite orbit group, and let $b$ be an admissible integer starting-coordinate parameter. Consider three-quadrant walks starting…

Atkinson–Sack conjecture. The generating function for the enumeration of sortable permutations is rational for every value of $k$.

CMPP's conjecture. The generating function for $\Lambda$-admissible colored partitions is

For positive integers $n$, let $N_n(t)$ be the numerator polynomial in … and define … where $2^{[m]}$ is the power set of $[m]$. Symmetric-difference formula conjecture. For all po…

Conjectural formula for $G_4$. One has

Berkovich–Uncu conjecture. The series $H_{L,s,k}(q)$ is eventually positive for all positive integers $L$, $s$, and $k$ satisfying $L\geq 3$ and $k\geq s+1$.

Let the classes be permutation classes defined by avoiding two patterns of length $4$ and one pattern of length $6$. Egge's conjecture. All of these classes are conjectured to be c…

Non-algebraicity conjecture. This latter generating series is not algebraic.

Andrews's conjecture. Every linked partition ideal has a two-variable generating function of the form

Let $k>3$. The generating function of $k$-noncrossing partitions is the formal power series that enumerates $k$-noncrossing partitions by their size. Bousquet-Mélou and Xin's conje…

The parametric algebraic-form conjecture.

Let $a$ and $b$ be positive integers with $\gcd(a,b)=1$, and let … Let $f$ denote the generating function for excursions, meanders, or bridges with step set $\mathcal{S}$. Minimal-…

Let $a_n$ be the number of triangulations of a fixed convex $n$-gon such that at least six triangles meet at each internal vertex. Let…

Let … where $\beta_{m,u}$ counts the relevant chinese-character diagrams of degree $m$ and loop number $u$. Pseudopolynomial conjecture. The Taylor coefficients of $b(x,y)$ in $x$…

Let … where $i$ is an integer, and let $s$ be the unique formal power series in $x$ and $p$ satisfying … For the Hurwitz generating series $H_g(x,p)$, and for a partition $\theta$…

For each $m\geq3$, let $P^{(m)}$ be the higher-point generating series, expressed using $w_1,\ldots,w_m$. Rationality and denominator conjecture. For each $m\geq3$, $P^{(m)}$ is a…