67 problems

Let a word be generated by an irreducible, doubly-stochastic, cyclic Markov chain on an alphabet of size $m$. Let its associated RSK Young-diagram shape be suitably centered and no…

Let $\rho$ be the uniform distribution on the diamond … Let $\Phi'$ denote the limit shape of the Young diagram associated with the corresponding locally uniform random permutation…

Krattenthaler–Müller's two-row formula conjecture. The average case complexity is

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…

Let $\lambda$ range over all partition shapes with $n$ cells. For a standard Young tableau of shape $\lambda$, define its sign by reading entries row by row from left to right, sta…

Little's conjecture. The two tableaux agree:

Let $\lambda$ be a partition with $n$ outside corners. Choose a corner $k$ and corresponding weight $\omega_\ell$, and retain the notions of Theorem … 1leq i1<cdots<irleq kleq jr<c…

Let $r$ be rank conditions satisfying the paper's rank-condition hypotheses, let $R_{ij}$ be the associated array of rectangles, and choose a fixed tableau $T_{ij}$ on each…

Let $T$ be a Richardson tableau, and let the order on Richardson components be given together with the operations ${\mathcal T}_{\alpha,\beta}$ and transposition. Richardson-compon…

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, let $I_T$ be the primitive ideal attached to $T$, let ${\mathcal V}_T$ be its orbital variety, and let…

For ${\mathfrak g}={\mathfrak s}{\mathfrak l}_n$, let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes. To each $T\in{\bf T}_n$ attach a primitive ideal $I_T$ and a…

Let ${\bf T}_n$ be the set of standard Young tableaux with $n$ boxes, and consider the combinatorics of inclusions among primitive ideals and among closures of orbital varieties at…

Let ${\mathfrak g}={\mathfrak s}{\mathfrak l}_n$, and let the algebraic and geometric orders be the orders on standard Young tableaux induced respectively by inclusion of primitive…

Let $k, ell\ge 0$ and $\beta,z>0$. Define … and let $r_{(k,\ell),\beta}(z)$ be the corresponding limiting integral ratio. Second-term oscillation conjecture. As $n\to\infty$, the q…

Octahedral-map conjecture. The map $\varsigma$ and the map defined in KTW are identical and linearly equivalent to maps in Theorem 1. The paper has already shown that its tableau-s…

Let $\varphi$ denote the RSK map and let $\phi$ denote the Littlewood–Robinson map; the source has previously established that both maps are one-to-one. Inverse-equivalence conject…

The Fundamental Symmetry maps are defined on Littlewood–Richardson tableaux: $\rho_1$ is the first Fundamental Symmetry map, $\rho_2$ is the map defined earlier in the paper,…

Let $c_\mu(r)$ be a cohomological quiver coefficient, and represent the relevant combinatorial data by a tableaux diagram: a sequence of chosen semistandard Young tableaux together…

Let $N\geq 3$ and let $\Omega_N(\mathbb R^D)$ denote the $N$-complex of tensor fields on $\mathbb R^D$ with the generalized differential. For $m\geq N$, consider the construction t…

Let $\lambda=(\lambda_1,\ldots,\lambda_r)\in\Lambda_m$ be a Young diagram with $\lambda_1$ boxes in its first row, and let $\lambda^{(n)}$ be obtained by adding a first row of $n$…

Let $\Delta^{k,m}$ be the star graph with $k$ branches, each receiving $m$ labeled chips, and consider the random labeled chip-firing process in which every firable vertex is equal…

Under the Robinson–Schensted–Knuth correspondence, a permutation is mapped to a pair of Young tableaux of the same shape. Let $A$ be the set of permutations whose tableaux avoid th…

Let $\lambda$ be a partition, and let $\mathrm{Skel}(\lambda)$ denote the skeleton associated with a connected crystal whose associated Young tableaux have shape $\lambda$. Let the…

Let $k\geq 1$ and let $T\in\operatorname{YT}(n)$, where $n\geq k+4$. Here $\operatorname{mM}_k(T)$ denotes the multiset of $k$-minors of the standard Young tableau $T$. Multiset re…