6 problems

Let $A = \{a_1,a_2,\dots,a_n\}$ be a set of positive integers with $a_1 < a_2 < \dots < a_n$. A Skolem set is a set for which $\{1,\dots,2n\}$ can be partitioned into pairs whose d…

Let $S_n$ be the symmetric group, and let a displacement pattern be a tuple $\alpha=(a_1,a_2,\dots,a_n)$ recording the displacements of a permutation, with the entries distinct. Th…

A perfect Skolem-type sequence of order $n$ is a sequence constructed from an ordinary perfect Skolem set of cardinality $n$. The $4k+1$ conjecture. For every order $n$, the number…

Let $A=\{a_1,a_2,\dots,a_n\}$ be a set with $a_1>a_2>\dots>a_n$ and total sum $\sum_{i=1}^n a_i=n^2$. A perfect extremal Skolem set is a perfect Skolem set satisfying this extremal…

Let $A\subset\{1,2,\dots,n\}$ have cardinality $n-2$, with exactly two missing elements $a_i,a_j\in\{1,2,\dots,n\}$. The two-missing-elements perfect Skolem set conjecture. The set…

Let $S_{2n}$ be the symmetric group, and let $l_t(\pi)$ denote the Toeplitz characteristic of a permutation. A fixed-point-free involution is a permutation $\pi \in S_{2n}$ satisfy…