14 problems

Fulman's semisimple-orbit measure conjecture. If the characteristic is a prime which is good and regular for $G$, choose $c$ uniformly among the $q^r$ $F$-rational semisimple orbit…

Let $D_1=\alpha\beta\gamma$ and $D_2=\alpha\beta^*\gamma$ be sequences of length $2n$ containing $n$ $1$s and $n$ $2$s. Call $D_1$ and $D_2$ $R$-related when $\beta$ contains equal…

Let $D_1$ and $D_2$ be decks, and let the descent polynomial of $\Pi(D_1;D_2)$ encode the numbers of permutations having each possible number of descents. Two-label counting-comple…

Conjecture. A similar result to Theorem $$ will hold for the descents of top-to-random.

Clay's position-matrix eigenvalue conjectures. The following assertions hold:

Multi-stack and restricted shuffle cutoff conjecture. Both chains exhibit cutoff, with the multi-stack chain around time

Let $H_{2m}$ be the $2m$th harmonic number, and let $F^{\mathbf{G}}(n,m)$ denote the expected number of correct guesses made by the strategy $\mathbf{G}$ for a deck of $n$ cards su…

Let $M$ be the position matrix of the one-shelf shuffling machine. Its eigenvalues describe the associated Markov chain on card positions; the previously established eigenvalues in…

The quadratic eigenvector conjecture. $M$ has an eigenvalue-eigenvector pair $(\frac{1}{4},\vec{v})$ with multiplicity at least $1$. Numerical computation suggests this eigenvector…

Single-directional CAT cutoff conjecture. The single-directional CAT shuffle should exhibit cutoff at

Let $\mathbf{X}^{(\mathrm{CTRT})}$ denote the Markov chain generated by the Cyclic-to-Random Transpositions shuffle, and let $\tau^{\mathrm{sep}}_{\mathrm{mix}}$ denote its separat…

No-better-dealing-method conjecture. Although other dealing methods may have the same coefficient of $a^{-1}$ for every value of $b$, there is no dealing method better than the pro…

Optimal dealing-method conjecture. The best dealing method is a method in which two of the sums equal $344$ and the other two equal $345$. This conjecture concerns minimising the l…

There are $n$ cards arranged on a circle. At each step, a nonempty interval of cards of length at most $L$ is chosen uniformly at random and its order is reversed. The $L$-reversal…