6 problems

Let $W(\ell_a,\ell_a)$ be the set of binary words containing $\ell_a$ occurrences of each letter, and consider its $\mathcal{UT}_2(\mathbb{T})$ equivalence classes. Generalised Cat…

Fix $r\in(0,1)$. For binary words in $W(\ell r,\ell(1-r))$, let $e(\ell,r)$ be the ratio of the number of $\mathcal{UT}_2(\mathbb{T})$ isoterms to the number of…

Fix $r\in(0,1)$ and consider words in $W(\ell r,\ell(1-r))$. Choose $w$ uniformly from this set, choose $u$ uniformly from the $\sim_2$-class of $w$, and choose $v$ uniformly from…

Let $W(\ell r,\ell(1-r))$ denote the set of binary words with the indicated numbers of the two letters. Fix $r\in(0,1)$ and $n\geq 2$, and let $i(n,\ell,r)$ be the fraction of…

Let $n\geq 2$. For a word $w$ over a two-letter alphabet, call $w$ locally isolated in $\mathcal{UT}_n(\mathbb{T})$ if every word $w'$ satisfying $w'\leftrightarrow w$ is inequival…

Let $\Sigma$ be a two-letter alphabet, let $w,v\in\Sigma^+$ with $w\neq v$, and let $n\geq 2$. Write $w\leftrightarrow v$ when $w$ and $v$ differ by one adjacent swap, and let…