19 problems

Let $G(q)$ be the $q$-series … and let $e(n)$ and $o(n)$ denote respectively the numbers of partitions of $n$ into odd parts whose index is even and odd. Coll–Mayers–Mayers conject…

Coll et al.'s nonexistence conjecture. No general linear greatest common divisor formula exists for the index of seaweed algebras when there are more than three parts in the top co…

Index formula conjecture. For all subsets $S$ and $T$ of $\Pi$,

Tauvel–Yu index formula conjecture. The upper bound is in fact an exact formula:

Let ${\mathfrak g}={\mathsf{Lie\,}}G$ be a simple Lie algebra, let $\mathfrak p_1,\mathfrak p_2\subset\mathfrak g$ be parabolic subalgebras with…

Let $\mathfrak{g}$ be a Frobenius maximal parabolic type-A seaweed, meaning a Frobenius seaweed subalgebra of $\mathfrak{sl}(n)$ obtained from a maximal parabolic construction. Its…

Recursive spectrum conjecture. The distinct eigenvalues in the spectrum of $\mathfrak{g}_{k,r}$ are precisely the integers in $[-k+1,k]$, and $\mathfrak{g}_{k,r}$ has the log-conca…

Parity-dependent spectrum conjecture. The seaweed $\mathfrak{g}$ is Frobenius, and the set of distinct eigenvalues in its spectrum is the collection of integers in

Stabilization conjecture. In either case, $\mathfrak{g}'$ is a Frobenius type-A seaweed with spectrum $S\cup S'$, where $S'$ is a multiset consisting of values contained in $S$; mo…

Let $\mathfrak{g}$ be a Frobenius seaweed subalgebra of $\mathfrak{sl}(n)$. Its spectrum is the multiset of eigenvalues of the adjoint action of a principal element of…

Let $p=(\lambda_1,\ldots,\lambda_l)$ be a partition of $n$ into odd parts, and let $\operatorname{ind}_S(p)$ denote its associated seaweed-index statistic. Parity conjecture. The i…

Let $C_{4,4}(n,k)$ be the composition-counting function. Red-cell recurrence conjecture. … This conjecturally describes the red cells in Table 4; no proof or resolution is supplied…

Let $C_{3,3}(n,k)$ be the composition-counting function and let $n_t$ denote the sequence associated with the displayed diagonals. Alternating-diagonal conjecture. … The formula re…

Let $C_{2,t}(n,k)$ be the composition-counting function. Second-diagonal conjecture. If $t$ is even, then … for integers $t>2$ and $s\geq 0$. This gives another observed diagonal r…

Let $C_{2,t}(n,k)$ be the composition-counting function. First-diagonal conjecture. If $t$ is even, then … for integers $t>2$ and $s\geq 0$. This describes one of the first nonzero…

Let $C_{p,q}(n,k)$ be the composition-counting function, and let $n_{1,k,t}$ and $n_{2,k,t}$ be the quantities appearing in the formula. Affine-formula conjecture. … for integers…

Let $C_{p,q}(n,k)$ denote the composition-counting function studied in the paper. Vanishing conjecture. … for integers $s,p,k\geq 0$ with $k=2^p$. The values of $n_{k,4}$ suggest t…

Let $A$ be any top or bottom block in a Frobenius seaweed of $\mathfrak{sl}(n)$ with $n\geq 2$. For each eigenvalue $i$, let $d_i$ be its multiplicity in the multiset of eigenvalue…

For a composition $\mathbf{a}=(a_1,\dots,a_r)\in\mathcal{C}_n$, let $p(\mathbf{a})=r$ be its number of parts, and let $F_{n,p}$ denote the number of Frobenius standard parabolic su…