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Disclaimer: I am not a professional graph theorist. Motivation: Let's consider the set $G_N$ of graphs with $N$ vertices where the vertices are assumed to be distinguishable. This set may correspon …

Motivation: This is related to a different question I asked in April. It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case …

Let's suppose $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$. We may then define: \begin{equation} S_n = \sum_{i=1}^n a_i \tag{1} \end{equation} Now, in order to estimate $\lvert …

I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $: …

It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality: \begin{equation} \lvert G_N \rvert = 2^{N \choose 2} \tag{1} \end{equation} Recently, I wondered how much bigger $\lv …

It turns out that my hypothesis (4) was completely wrong. Besides Example II.15 in Flajolet and Sedgewick, Analytic Combinatorics pointed out by Richard Stanley, another useful reference is The Asympt …

Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct …

Might there be a probability distribution $\mathcal{D}$ such that if we sample $a_i \sim \mathcal{D}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$ then if we define the asymptotic estimate $f$: \begin{ …

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be u …

Motivation: Let's define a function computable on a $k$-ary tree as a function composed with simpler computable functions defined at each node such that a function of this kind defined on a binary tre …