8 problems

Major-index conjecture. This expected value is a rational function in $n$ that is $1/(k+1)$ times the quotient of a monic degree-$(k+1)$ polynomial by a monic degree-$k$ polynomial…

General skew-shape asymptotic normality conjecture. Asymptotic normality for general skew shapes, rather than only block diagonal skew shapes, holds if and only if

Log-concavity conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are almost always log-concave for partitions $\lambda\vdash n$ for large $n$.

Near-unimodality conjecture. The polynomials $\SYT(\lambda)^{\maj}(q)$ are nearly unimodal but not unimodal for partitions $\lambda$ or $\lambda'$ in the following cases: any parti…

Conjectured three-rowed formulas. The two displayed identities hold.

Let $\mathfrak i_j$ denote the increasing permutation of length $j$, let $\mathfrak d_j$ denote the decreasing permutation of length $j$, and let $12[\alpha,\beta]$ denote the dire…

Let $EN_{s,t}(\sigma)[q]$ and $NE_{s,t}(\sigma)[q]$ denote the major-index generating functions for the indicated pattern-avoiding linear extensions of rectangular posets. Let…

Let $\mathcal{S}_n$ be the set of permutations of $[n]$, and let $\mathcal{S}_n(\tau)$ denote the permutations avoiding the pattern $\tau$. For a permutation $\pi$, its major index…