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Gabor Toth's Glimpses of Algebra and Geometry contains the following beautiful proof (perhaps I should say "interpretation") of the formula $\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} …

The reason this is called the "exp-log" correspondence is that it can be written in terms of formal power series as follows: write $G(z) = \sum g_n z^n, F(z) = \sum f_n z^n$. We need the additional hy …

$\newcommand\Schur{\mathrm{Schur}}\newcommand\op{^\text{op}}\newcommand\FinVect{\mathrm{FinVect}}\DeclareMathOperator\Rep{Rep}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\GL{GL}\DeclareMathOper …

Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell i …

For example, according to Stanley the identity $n \cdot \text{pp}(n) = \sum_{i=1}^{n} \sigma_2(i) \text{pp}(n-i)$ has no known bijective proof, where $\text{pp}(n)$ denotes the number of plane partiti …

This is the Hankel determinant associated to the sequence $m_n = \mathbb{E}(X^n) = n!$ of moments of an exponential distribution with mean $1$. Some general results can be used to show that the sequen …

I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of t …

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n …

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As …

Here is a more informative version of this identity. Let $Z_n$ denote the cycle index polynomial of the symmetric group $S_n$, namely $$Z_n = \frac{1}{n!} \sum_{\sigma \in S_n} z_1^{c_1(\sigma)} z_2^{ …

The Bell numbers satisfy $\frac{\ln B_n}{n} \sim \ln n$ which is faster than exponential, so the ordinary generating function $\sum B_n x^n$ has zero radius of convergence. As a more elementary argume …

I'll rename all three of your variables; you are asking for the number of partitions of $k$ that fit into an $m \times n$ box. This is famously known to be the coefficient of $q^k$ in the $q$-binomial …

Let $L$ be a language on a finite alphabet and let $L_n$ be the number of words of length $n$. Let $f_L(x) = \sum_{n \ge 0} L_n x^n$. The following are well-known: If $L$ is regular, then $f_L$ is …

Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have $$q^{\frac{2}{27} n^3 + O(n^{8/3} …

Expanding on Richard's comment: let me rename your graph to $S$ and consider the adjacency matrix $A$ abstractly as a linear operator acting on the free vector space $\mathbb{C}[S]$ on (the vertices o …