Recall that in the $n$-simplex $\Delta[n]$, we have a combinatorially crucial bijection between facets, (codimension $1$ faces) and vertices, where the $i$th face of a simplex corr …

Statement of Van Den Berg-Kesten-Reimer inequality: Let $n$ be a positive integer. For $i\in[n]$, let $\mu_i$ be a probability measure on a finite set $\Omega_i$. Let $\Omega=\Ome …

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs: let's call two eta product identities $\sum\ …

First we define a function $f(x,p)$, with $x$ a natural number and with $p$ a prime number. $f(x,p)$ stands for the location where the digit "$p-1$" first appears in the base-$p$ …

Is there a characterization of graphs $G$ such that $\exists$ $\phi : G \rightarrow KG(n,k)$, where $KG(n,k)$ is the Kneser graph ($k \leq \lceil \frac{n}{2}\rceil $). Any refe …

Consider a minimally connected graph (i.e., a spanning tree) on $n$ nodes, $\mathcal{T}=(\mathcal{V},\mathcal{E}_{\tau})$, and its complement $\overline{\mathcal{T}}=(\mathcal{V}, …

Again, there is a general and a concrete question: Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N …

The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows. Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ ar …

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$? I've listed out the first few terms: for $x=0,1,2,3,4,5 …

Take a naive interpretation of regular polyhedra: All vertices (including epsilon ball) congruent All edges congruent All faces congruent We can now find interesting families b …

What is the probability of two permutations on set X of size m (i.e. |X|=m) having at least n points of intersection? By this I mean that if two permutations, which I'll call g(x) …

Hello everyone. I'm really struggling with this question. All help appreciated. Find the minimum positive integer r for which there exists an r-regular graph G such that λ(G) ≥ κ( …

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is Möbius nearly Orthogonal to Morse ! Harold Calvin Marsto …

This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest possible proof strategies? I have vectors $v_1, \ldots, v_k$ in ${\bf R}^n$. Each of t …

Hello, can anyone recommend good combinatorics textbooks for undergraduates? I will be teaching a 10-week course on the subject at Stanford, and I assume that the students will be …