9 problems

For each positive integer $k$, let $h_k(x)$ and $f_k(x)$ be the polynomials associated with $312$-avoiding reduced valid hook configurations, with $h_k(x)=f_k(x-1)$. Sankar's conje…

Let $\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))$ denote the set of $312$-avoiding reduced valid hook configurations with $k$ hooks on permutations with $2k+i$ points. D…

Let $\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))$ denote the set of $312$-avoiding reduced valid hook configurations with $k$ hooks on permutations with $2k+i$ points. D…

Let $\operatorname{RedVHC}_k(\operatorname{Av}_{3k-i}(312))$ denote the set of $312$-avoiding reduced valid hook configurations with $k$ hooks on permutations with $2k+i$ points. D…

For $n\geq 3$, let $\operatorname{VHC}(S_{n-1})$ denote the set of valid hook configurations of permutations in $S_{n-1}$. Oddness conjecture. If $n\geq 3$, then…

Weak Bruhat monotonicity conjecture. If $\sigma,\tau\in S_r$ satisfy $\sigma\leq_B\tau$, then

Alternating-sum Catalan conjecture. For every $k\geq 1$,

Three-dimensional Catalan conjecture. For every $k\geq 1$,

Let $\operatorname{Av}_n(\sigma_1,\ldots,\sigma_r)$ denote the permutations of length $n$ avoiding the listed patterns, and let $\operatorname{\mathsf{VHC}}(\mathcal{S})$ denote th…