14 problems

Dokos et al.'s conjecture. The patterns $3142$, $3241$, and $4132$ are $\operatorname{maj}$-Wilf equivalent, and the patterns $1423$, $2314$, and $2413$ are $\operatorname{maj}$-Wi…

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…

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…

Let $\operatorname{Maj}(n,q)$ be the random permutation produced by the major-index sampler with parameter $q$, and let $\operatorname{LIS}(\operatorname{Maj}(n,q))$ denote its lon…

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…

Local limit conjecture. For every partition $clambda cvdash n$, uniformly in $n$ and over all integers $k$,

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

Local limit conjecture. Uniformly for all $n$ and all integers $k$,

Conjectured related-shape formulas. All three displayed identities hold.

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

For a permutation $\sigma$, let $\operatorname{Des}(\sigma)=\{i:\sigma_i>\sigma_{i+1},\ 1\leq i\leq n-1\}$, and define … An involution is a permutation satisfying…

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 $\operatorname{Av}_n(321)$ denote the permutations of length $n$ avoiding the pattern $321$, and let the major-index polynomial sequences be the corresponding generating polyno…