11 problems

Let $[n]=\{1,\ldots,n\}$, and let a Universal Cycle for $t$-multisets of $[n]$ be a cyclic sequence of $\binom{n+t-1}{t}$ integers from $[n]$ in which every $t$-multiset of $[n]$ a…

Let $n,t\geq 2$ and let $w<t$. Let $\mathcal{V}_t(n,w)$ be the universal cycle constructed from the relevant fixed-weight strings, and let $\mathcal{V'}_t(n,w)$ be the sequence obt…

Uniform-slicing conjecture. For every $0\leq i\leq n-d$, if the slicing length is $k^i|u|$, then the resulting collection is an upfamily.

Let a $(v,K,\lambda)$-pairwise balanced design be a $v$-vertex hypergraph whose $2$-degree is $\lambda$ and whose edge cardinalities belong to a set of integers $K$. A base block i…

Let $S_n$ be the set of permutations of $[n]$, and let $\mathcal{O}(n)$ be the clustered graph of overlapping $n$-permutations: its edges are the permutations in $S_n$, and consecu…

Let a gucycle be a cyclic ordering whose windows represent each isomorphism class of graphs on $n$ vertices exactly once. Corrected gucycle conjecture. For each $n\geq 4$, there ex…

Let a gucycle be a cyclic ordering whose windows represent each isomorphism class of graphs on $n$ vertices exactly once. Brockman–Kay–Snively's conjecture. For each $n\neq 2$, the…

Chung–Diaconis–Graham conjecture. For every $k\in\mathbb{N}$, there exists $n_0\in\mathbb{N}$ such that for all $n\ge n_0$, there exists a universal cycle for $\binom{[n]}{k}$ when…

Let $\textbf{T}(n,k)$ denote the set of strings of length $n$ over the paper's alphabet, and let $\textbf{S}\subseteq\textbf{T}(n,k)$ be closed under rotations. Given a string…

Let $\textbf{T}(n,k)$ denote the set of strings of length $n$ over the alphabet of symbols used in the paper. Let $\gamma\in\textbf{T}(m,k)$ be a string containing a symbol…

Let $k$ be a positive integer. A U-cycle of isomorphism classes of graphs on $k$ nodes is a cyclic sequence in which every $k$-window represents a distinct isomorphism class of gra…