Mathematics Colloquium Date: Wednesday, 30 August 2023 Time: 3:30  4:30 PM Venue: Seminar Hall A probabilistic proof of two combinatorial identities Patrick Polo IMJPRG, Université Pierre et Marie Curie. 300823 Abstract This talk aims at telling several stories, mathematical or otherwise. In particular, let $X$ be a denumerable probability space, $n$ an integer $\geq 2$, endow $X^n$ with the product probability. The computation of the probability of the subset $X(n)$ of $n$tuples of pairwise distinct elements is part of the folklore in statistical mechanics and in combinatorics, but does not seem to be as wellknown as it could be. We give an elementary proof for a slightly more general computation, and obtain along the way that for each positive integer $n$, the sum of the terms $(1)^k c_k(n)$, where $c_k(n)$ denotes the number of connected graphs with $n$ vertices and $k$ edges equals $(1)^{n1} (n1)!$.(There is another proof using generatingfunctionology).
