105 problems

For all finite trees $T_1$ and $T_2$, if their chromatic symmetric functions are equal, $X_{T_1}=X_{T_2}$, then the trees are isomorphic, $T_1\cong T_2$. Here, for a finite graph…

For every finite preorder $\tau$, the $h$-polynomial of $\tau$ equals that of its dual preorder: $h_{\tau}(t)=h_{\tau^{*}}(t)$, where $\tau^{*}$ denotes the dual preorder.

The conjectures concern permutation statistics on $\mathfrak{S}_n$. First, the generating function formed from…

For every finite simple graph $G$ with distinguished vertex $v$, let $\operatorname{Aut}_v(G)\leq \operatorname{Aut}(G)$ be the subgroup fixing $v$. Then the conjecture asserts tha…

For every finite Coxeter group $W$ of rank $r$, the $h$-polynomial of the order complex of its noncrossing-partition lattice admits a nonnegative real-rooted symmetric decompositio…

Conjecture 9.1 proposes the identity … where $\mathsf{LD}(0,n)^{\ast k,\circ r}$ is the set of positively labelled Dyck paths of size $n$ with $k$ decorated rises and $r$ decorated…

Conjecture 2 (Brenti). Let $W$ be a finite Coxeter group. For $u,v\in W$ with $u\leq v$ in Bruhat order, define $\widetilde{R}_{u,v}$ by

Let $m,n$ be positive integers, let $L_{mn,n}^+$ denote the set of rational $m$-Dyck paths, and let $L_{n,n}^+$ denote the set of Dyck paths of size $n$. For any rational Dyck path…

Let $X_w$ be a Schubert variety, and let $X_v$ range over the Schubert varieties contained in its maximal singular locus. The maximal non-Gorenstein locus characterization proposes…

Let $\tau$ be a linear fractional transformation of order $2$, and let a bigrid be an array with entries $\gamma(\epsilon,i,j)$ wherever they are defined. Let a bigrid be perfect i…

Let a bigrid be an array with entries $\gamma(\epsilon,i,j)$, where the indices are taken wherever the entries are defined, and let a bigrid be perfect in the sense defined in the…

Following the standard shelling of the order complex of the truncated Boolean algebra $B_n-\{\hat{0},\hat{1}\}$, let $\Delta_n$ be the resulting $h$-complex. Edelman–Reiner conject…

Uyemura-Reyes's conjecture. The eigenvalues of $\upsilon_n$ are rational integers.

Let $\lambda$ be a partition with $\ell$ parts and fixed parity for each part. Let $G_\lambda(x/x)$ be the specialized symmetric function defined by substituting $2p_{2i-1}$ for…

Divided-differential-operator dimension conjecture. One has

Let $\mathcal{A}$ and $\mathcal{B}$ be real central hyperplane arrangements, and let $R$ be an integral domain with $\operatorname{char}R\neq 2$. Write…

Kirillov's conjecture. $f_{u,v,w}(N)$ is eventually polynomial in $N$.

Let $\mathbb{F}$ be a field, let $M$ be a nonsingular matrix over $\mathbb{F}$, and let $x$ be a vector over $\mathbb{F}$. Alon–Jaeger–Tarsi conjecture. For any field $\mathbb{F}$…

Let $L$ be a Latin square of order $n$, and define its sign by the product of the signs of its row and column permutations. Let $E_n$ and $O_n$ denote the numbers of even and odd L…

Class-number conjecture. If

Let $B_n$ be the signed symmetric group on $n$ letters, with inclusion $B_n\subset B_{n+1}$ given by choosing the elements of $B_{n+1}$ that fix the $(n+1)$st letter. For…

Let $m \geq 2$ be a fixed integer. A geometric distance-regular graph has diameter $D \geq 3$ and intersection number $c_2 \geq 2$. Koolen–Bang's classification conjecture. Any geo…

Let $m \geq 2$ be a fixed integer. A distance-regular graph is coconnected if it is not a disjoint union of complete graphs. Let $\phi_1$ and $\tau_2$ be the clique parameters asso…

Let $\Gamma$ be a geometric distance-regular graph with diameter $D \geq 3$ and distinct eigenvalues … Let $b = \frac{b_1}{\theta_1 + 1}$. For vertices $x,y$ at distance $i$, let…

Let $\Gamma$ be a geometric distance-regular graph with diameter $D$ and distinct eigenvalues … Let $b = \frac{b_1}{\theta_1 + 1}$. For a Delsarte clique, let $\phi_i$ denote the n…