23 problems

Let $[n]=\{1,2,\ldots,n\}$, and let $\mathbf{S}_k(n)$ denote the set of $k$-subsets of $[n]$. A standard representation represents each subset by an ordering of its elements. When…

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…

Let $m$ and $k$ be positive integers. An $(m,k)$-Ucycle is an $m$-distinguishable sequence in which no $m$-window repeats an element and every $m$-subset of $[k]$ occurs exactly on…

Let $m$ and $k$ be positive integers. An $(m,k)$-Mcycle is a cyclic $m$-distinguishable sequence on $[k]$ in which every $m$-multiset of $[k]$ appears exactly once; it therefore ha…

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.

Upfamily-from-alphabet-multiplier conjecture. There exists at least one upfamily $\mathcal{F}$ constructed by slicing $v$ into identically sized elements.

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…

An upword for $\{0,1\}^n$ is a word containing every binary word of length $n$ as a consecutive factor, with the symbol $\mathord{\diamond}$ permitted as a wildcard according to th…

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…

Let $G$ be the cluster graph for $n$-permutations: its vertices are the order-isomorphism classes of the first $n-1$ entries of permutations, and its edges are the $n$-permutations…

Let $S_n$ be the set of permutations of $[n]$. A universal word for $S_n$ is a word over $\mathbb{N}$ whose consecutive substrings of length $n$ represent each permutation in $S_n$…

Let $S_n$ be the set of permutations of $[n]=\{1,2,\ldots,n\}$. A word with incomparable elements covers every permutation that is a linear extension of the order it specifies. Usi…

Chung et al.'s conjecture. For each $k$, there exists an $n_0(k)$ such that ucycles exist for $k$-subsets of $[n]$ provided that $n\ge n_0(k)$ and the divisibility condition…

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…

Shortening conjecture for u-cycles. Using incomparable elements at distance $n-1$, one can obtain u-cycles for $n$-permutations of lengths

Hurlbert et al.'s conjecture. A universal cycle exists for all $k$-multisets of $[n]$ provided that $n$ is sufficiently large and

The universal-cycle existence conjecture. For some $n$ sufficiently large, $A$ has a universal cycle.

The u-cycle conjecture. The collection of all $k$-subsets of $[n]$ has a universal cycle provided that

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…