5 problems

Let $S_n$ be the symmetric group, and let $f\colon S_n\to\{0,1\}$ be a Boolean function. Say that $f$ is $\epsilon$-close to degree $d$ if its distance from the space of degree-$d$…

Let $S_n$ be the symmetric group, and let $fa S_n\to\{0,1\}$ be a Boolean function. Say that $f$ is $\epsilon$-close to degree $d$ if there exists a degree-$d$ function…

Let $n$ be a positive integer, let $\mu\vDash n$ be a partition of $n$, and let $\operatorname{Cr}(\mu)$ denote the corresponding conjugacy class in $S(n)$. Let…

Let $S(n)$ be the symmetric group equipped with the metric $\mathrm{d}$, and let $A,B\subseteq S(n)$ be nonempty. Write $M$ for their midpoint set, and let $\mathrm{d}(A,B)$ denote…

Let $I_n$ be the set of involutions in the symmetric group $S_n$, and let $F_n^a$ denote the set of involutions with exactly $a$ fixed points. In particular, write $F_n^0$ and…