All Questions

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 ...

Famously, the alternating sign matrix theorem gives a product formula for the number $a(n)$ of ASMs of size $n$. There are multiple proofs of this formula, all somewhat involved. My question is ...

Definition Call a partition $\lambda$ of an even integer $2n$ "black-white balanced" if the following equivalent conditions are satisfied: In the usual (Ferrers-)Young diagram of $\lambda$, ...

I am looking for natural bijections from the set of integer partitions of $n$ to itself. Of course, I have no definition of natural, but for the purpose of this question it suffices that it appears ...

A current project uses bijections from a set to itself. (The set is the integer compositions of $n$, i.e., "ordered partitions of $n$," but that doesn't seem pertinent to the question.) Is ...

A partition of $n$ is a unordered list, whose sum is exactly $n$. The total number of $2$’s in all partitions of $n$ is equal to the total number of singletons in all partitions of $n−1$. A singleton ...