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 IMJ-PRG, Université Pierre et Marie Curie. 30-08-23 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 well-known 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 $(-1)^{n-1} (n-1)!$.(There is another proof using generatingfunctionology).
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