100 problems

Let $e$ be a prime, let $n$ be a positive integer, and let $\lambda$ be an $e$-regular partition of $n$. Over a field of characteristic $e$, write $D^{\lambda}$ for the irreducible…

Oblak's conjecture. The size-preserving function $P\mapsto\operatorname{Obk}(f(P))$ is precisely $\mathfrak{D}$.

Fomin–Fulton–Li–Poon's conjecture. The difference

Degree bound conjecture. The degree bound referred to in the paper is not optimal, and

Box Conjecture. The elements of $\mathcal{D}^{-1}(Q)$ can be arranged in this box so that the partition in position $(i_1,i_2,\cdots,i_k)$ has exactly

Let $\overline{\zeta}_n$ be the symmetrized Mordell–Tornheim zeta value of order $n$, and let $p(n)$ denote the partition function. A monomial has total weight $n$ when the sum of…

Let $\mathsf k$ be an infinite field, let $B$ be a nilpotent matrix, and let $\mathfrak D(P)$ denote the maximum Jordan partition arising from a nilpotent matrix commuting with one…

For a positive integer $n$, let a partition into distinct parts be counted according to whether it has more odd parts than even parts or more even parts than odd parts. Kim–Kim–Lov…

Let $n$ be a positive integer, let $\frac{r}{s}$ be a term of the $n$-th Farey sequence, and let $\tilde{M}_{r/s}$ and $\tilde{M}^{\prime}_{r/s}$ be the compositions of generalized…

Let $n$ be a prime number, let $\operatorname{sgn}_n$ be the partition of $n$ corresponding to the sign representation of $S_n$, and let $r$ be a term in the $n$-th Farey sequence.…

Han's conjecture. For every $k\in\mathbb{N}$, $P(n)$ is a polynomial in $n$.

Let $\lambda$ range over partitions of $n$. For a box $\square\in\lambda$, let $h_{\square}$ denote its hook length, and let $f_\lambda$ denote the number of standard Young tableau…

Chow–Long's conjecture. The set $S_\alpha$ is uniquely avoidable.

Square-root discrepancy conjecture. The best bound for this discrepancy, uniformly over all $N$ and all such vector sequences, is of order $\sqrt d$.

Let $n$ be a positive integer, let $partition \lambda\vdash n$ have length $\ell(\lambda)$ and rank $\operatorname{rank}(\lambda)$, and let $\hat{s}^2_\lambda$ denote its 2-bottom…

Let $lambda$ be a partition and let $rgeq 0$. For the Pieri coefficients $c_i^{\alpha}(lambda)$, define … Let $d_\rho(lambda)$ and $z_\rho$ denote the quantities associated with a…

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathcal{NP}_{\mathrm{2res}}(\theta)$ be the set of skew diagrams…

Let $\mathcal{P}$ be the set of partitions, let $\mathcal{P}_{\mathrm{2reg}}$ be the set of 2-regular partitions, and let…

Let $\mathcal C_k$ denote the limit shape of the random-core growth process for fixed $k$. Let $D_k$ be the piecewise-linear curve with vertices … where $\gamma$ is the scaling con…

Symmetry of complements. For all $\lambda\in\mathcal R_k$, $\pi(\lambda)=\pi(\lambda^{\mathrm{comp}})$. The conjecture is supported in the paper by computations for $k\leq 6$.

k-complementation symmetry. For every $\lambda\in\mathcal R_k$, $\pi(\lambda)=\pi(\lambda^{\mathrm{comp}})$.

Spanning-set reduction conjecture. The spanning set can be reduced by removing every vector $\alpha(-\lambda)X(-\mu)v_\Lambda$ for which the partition $\mu$ has a sub-partition, no…

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a partition, and define … For a partition $\lambda$, let $z_\lambda$ be the standard symmetric-function multiplicity factor;…

For each integer $n\geq 1$, let $z_\lambda$ be the standard symmetric-function multiplicity factor for a partition $\lambda$, and retain the notation $\operatorname{Ev}(\lambda)$,…