Chennai Mathematical Institute

Seminars




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).