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Let the (homogeneous) Gaussian numbers be defined as $[n]_{x,y} = \frac{x^n-y^n}{x-y}$, and define Gaussian factorials as $[n]_{x,y}! = [n]_{x,y}[n-1]_{x,y}\dots [1]_{x,y}$, and the Gaussian binomials …

Expanding on my comment, here's what the two steps look like combinatorially. Induction from $S_k$ to $B_k$ sends $S^\lambda$ to $\bigoplus_{\mu, \gamma} W(\mu,\gamma)^{\oplus c^\lambda_{\mu,\gamma}}$ …

Too long for a comment, but not completely thought out: Call a point $x$ a "bend" if $f(x-1) = f(x+1)$. I think if we restrict to the subclass of $C_2$ with finitely many bends then the only irreduci …

I am interested in a possible generalization of the following fact from combinatorics: If $n<m$ then there are at least as many ways to partition a set of size $nm$ into $m$ sets each of size $n$ as t …

The branching rule for hook partitions is multiplicity free and pretty easy to describe: $$s_{(k,1^j)} = \sum_{i < k/2}o_{(k-2i,1^j)} + \sum_{i < k/2}o_{(k-2i-1,1^{j-1})}$$ That is: You either remove …

Let $G$ be a graph embedded (without crossings) on a torus $T$. It's fairly well known that this implies the chromatic number of $G$ is at most 7. If I lift $G$ to the universal cover of $T$, we get a …

Let's look at the ratio of two adjacent $q$-binomials as we move away from the center, for simplicity I'll do the even case. $\binom{2n}{n-a}_q / \binom{2n}{n-a-1}_q = \frac{[n+a+1]_q}{[n-a]_q} > q^{2 …

I came across the following cute fact about partitions: \begin{align} & |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt] & {} - |\{ \lambda \vdash n \text{ with an odd number o …

Consider the polytope in $\mathbb{R}^3$ with $8$ vertices at coordinates $(\pm 1, \pm 2, 1), (\pm 2, \pm 1, -1)$. Geometrically this looks like a cube where the top face is stretched in the direction …