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I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example: In the case where the $4$...

Let $n,H$ two fixed positive integers. Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...