9 problems

Let $X$ be a finite proper subset of $\bigcup_{j\geq 2}\mathfrak{S}_j$, meaning that $X$ does not contain both an identity permutation $12\cdots j$ and a reverse identity permutati…

Logarithmic-length conjecture. For every integer $n\geqslant 1$, there exists a binary word $\mathfrak{w}$ such that

Three-quarter palindrome conjecture. The word $w$ has a palindromic subsequence of length at least

Orthogonal-cut conjecture. For every such $w$ and every such linear representation, the maximum of the lengths of $p_1p'_1$ and $p_2p'_2$ is at least

Strong circular-palindrome conjecture. There is such a partition with subsequences $s_1,s_2$ satisfying

Weak circular-palindrome conjecture. Every binary circular word of length $n$ has a palindromic subsequence of length at least

Brevier–Preissmann–Sebő's conjecture. The word $w$ can be partitioned into two linear words $w_1,w_2$ of equal length, $w=w_1w_2$, having subsequences $s_1,s_2$ such that $s_1s_2$…

Lyngsø and Pedersen's conjecture. Every binary circular word of length $n$ divisible by $6$ with equal numbers of zeros and ones has an antipalindromic subsequence of length at lea…

Let $\pi$ be uniformly distributed in ${\mathcal S}_n$, and let $L_{n,k}=L_{n,k}(\pi)$ denote the length of its longest $k$-alternating subsequence, where a subsequence is $k$-alte…