8 problems

Let $\mathfrak A_n$ denote the alternating group, let $f_n$ be the function defined by the paper's Algorithm 1, and let $\leq$ denote the Bruhat order on $\mathfrak A_n$. Bruhat-or…

Let $\mathbf{m}\in\mathbb{E}_n$, let $\lambda\vdash n$, and let $T\in\operatorname{HikSYT}(\mathbf{m},\lambda)$. For the recursively defined Hikita weight $h_T(q)$ and every real…

Let $\mathbf{m}\in\mathbb{E}_n$, let $P=P_{\mathbf{m}}$ be the associated natural unit interval order, and let $\lambda\vdash n$. Let $\operatorname{strongST}_P(\lambda)$ be the st…

Let $P$ be a natural unit interval order with $n$ elements, let $\lambda\vdash n$, and let $\operatorname{strongST}_P(\lambda)$ be the set of strong standard $P$-tableaux of shape…

Let $P$ be a unit interval order corresponding to an incomparability graph $G(d)$ and to a lex-maximal part listing encoded by ${\bf a}=(a_1,\ldots,a_n)$. Let $d'$ be the Dyck path…

Main conjecture. The claim of the stated theorem is true for all unit interval orders $\textbackslash \mathcal{P}$, except for its final assertion concerning equality of tableau-sh…

Column-reading conjecture. The decomposition of the class's symmetric-function image into Schur functions is given by the column-reading words of the $\textbackslash \mathcal{P}$-t…

The $\textbackslash \mathcal{P}$-Knuth Schur-positivity conjecture. The images of $\textbackslash \mathcal{P}$-Knuth equivalence classes under the standard map are symmetric and Sc…