5 problems

For $k\geq 1$, let $f(k)$ be the minimum $n$ such that there exists a permutation $\sigma\in S_n$ containing every permutation in $S_k$ as a pattern; such a permutation is called a…

Let $L_k$ denote the minimum length of a permutation containing every permutation pattern of length $k$. Arratia's conjecture. Asymptotically, … Arratia's conjecture concerns the a…

Let $L_k$ denote the minimum length of a permutation containing every permutation pattern of length $k$. Eriksson et al.'s conjecture. Asymptotically, … The cited 2007 work gives t…

For $k\geq 1$, let $[k]=\{1,\ldots,k\}$, and let a bi-directional circular pattern of a word $w\in [k]^n$ mean a permutation that is a pattern of some cyclic rotation of $w$ or of…

For $k\geq 1$ and an alphabet size $r$, let $f(k;r)$ be the minimum $n$ such that there is a word $\sigma\in [r]^n$ containing every permutation in $S_k$ as a pattern; such a word…