25 problems

Let $C>0$. A Young diagram $\lambda$ with $n$ boxes has at most $C\sqrt{n}$ rows and columns, and let $\pi\in S_n$ be a permutation. Moore and Russell's character bound. There exis…

Let $Y$ be the Young diagram under consideration, let $S_Y$ denote its transversals, and let $S_Y(213)$ denote the transversals avoiding the pattern $(213)$. For a transversal…

Let $\mathfrak{so}(N)$ be the orthogonal Lie algebra, and let $\operatorname{ad}$ denote its adjoint representation. For a Young diagram $\lambda$, write $\lambda\vdash 2m$ when it…

Let $Y$ be a Young diagram and let $H(Y)$ be the associated 3-partite 3-uniform hypergraph. Write $|Y|$ for the number of cells of $Y$, let $\tau^{(2)}(H(Y))$ denote its second cov…

Let $Y$ be a Young diagram, and let an allocation mean a coarse filling obtained by subdividing $Y$ into subrectangles and partitioning the symbol set into smaller subsets, with pr…

Let $Y$ be a Young diagram with row lengths $l_1,\ldots,l_m$. It is Latin if its cells can be assigned integers so that row $i$ contains $1,\ldots,l_i$, and the entries in each col…

McNamara–van Willigenburg's conjecture. The diagrams $\mathcal{D}$ and $\mathcal{E}$ satisfy $\mathcal{D}\sim\mathcal{E}$ if and only if the factors satisfy:…

For a hypergraph $H$, a 2-matching is a set of edges such that every two edges share fewer than two vertices, and let $\nu^{(2)}(H)$ denote its maximum size. For a Young diagram…

A Young diagram $Y$ is wide if, for every subset $S$ of its rows, the diagram formed by $S$ dominates its conjugate; a filling of $Y$ is Latin if it assigns to each row the numbers…

Let $n$ and $k$ be positive integers, let $\lambda$ range over all partitions contained in an $n\times k$ rectangle, and write $\overline{\lambda}$ for the complementary partition…

Let $r\geq 2$ and $1\leq i\leq r$. Let $\mathcal{B}_{r,i}$ be the set of partitions whose $(i-1)$-bottom dissection consists of $i-1$ successive bottom squares followed by bottom r…

Let $\lambda$ be a symmetric Young diagram that is not a hook, let $G_\lambda$ be the associated group, and let $\dim\lambda$ denote the dimension of the corresponding representati…

Let $G_\lambda$ be the group associated with a Young diagram $\lambda$, and let $\dim\lambda$ denote the dimension of the corresponding representation. For…

Let $G_\lambda$ be the group associated with a Young diagram $\lambda$, and let $\dim\lambda$ denote the dimension of the corresponding representation. Consider the diagrams…

Let two Young diagrams be related by adding one box to the first row, with the corresponding parameter in the interval $(0,1/n)$. Added good box conjecture. At every step of the ab…

Let $\pi$ be a $123$-avoiding permutation, and let its boundary grid be the associated skew Young diagram. The downcore graph has the boxes of the boundary grid as vertices, with a…

Let $\lambda=\mathbf{r}^{\mathbf{p}}$ be a Young diagram in multirectangular coordinates. For a partition $\mu$, let $J^{\star,(\alpha)}_\mu$ be the shifted Jack function and let…

Let $\lambda=(p_1,p_2)\times(q_1,q_2)$ be a multirectangular Young diagram consisting of at most two rectangles, and let $\operatorname{Ch}_\pi^{(\alpha)}(\lambda)$ and the orienta…

Let $b\in P_+$ correspond to a Young diagram $\mu_b$ with rows of lengths $m_i$, and let $b_i=m_i\omega_1$. Write $\operatorname{ord}$ for the order of $\mu_b$. Evaluation conjectu…

Let $M^{-1}$ be the inverse matrix indexed by paths, and let $\mu$ be a column index. For each chord $c$ under $\mu$, let its height be the number specified by the diagram, and let…

Let $M^{-1}$ be the inverse matrix indexed by paths, and let $\lambda$ be a row of order $n$. For a chord $c$ of $\lambda$, write $|c|$ for its length, and let $f_{\lambda/\mu}(q)$…

Let $\nu=(\mu_1^2,\dots,\mu_n^2)$ be a doubled partition with distinct parts, where $\mu_i^2$ denotes that the part $\mu_i$ occurs twice, and let $W_{\nu}(z)$ be its associated Wro…

Let $\lambda$ be a partition, and let $W_{\lambda}(z)$ denote the Wronskian of Hermite polynomials associated with $\lambda$. Simple-zero conjecture. For every partition $\lambda$,…

Let $\lambda$, $\mu$, and $\nu$ be Young diagrams with $|\lambda|+|\mu|=|\nu|$. Let $A$ be an admissible order on the skew Young diagram $\nu\setminus\lambda$, and let $A'$ be an a…

Let $\lambda,\mu,\nu$ be Young diagrams. Let $A$ be an admissible order on $\nu\setminus\lambda$ and let $A'$ be an admissible order on $\mu$. The set…