**Problem of the Week**

**BUGCAT**

**Zassenhaus Conference**

**Hilton Memorial Lecture**

**BingAWM**

**Math Club**

You are here: Homepage » Seminars - Academic year 2023-24 » Combinatorics Seminar » 2014 September 2

seminars:comb:comb_abs_20140902

**Matthias Beck (San Francisco State)**

**Permutation Descent Statistics via Polyhedral Geometry**

Abstract for the Combinatorics Seminar

Permutations are some of the most fundamental objects of mathematics. A basic combinatorial statistic of a permutation $\pi\in S_n$ is the number of descents, $des(\pi):=\#\{j:\pi(j) > \pi(j+1)\}$. Euler realized that $$ \sum_{k\geq 0}(k+1)^n t^k = \sum_{\pi\in S_n} t^{des(\pi)}/(1-t)^{n+1}$$ and there have been various generalizations of this identity, most notably when Sn gets replaced by another Coxeter group.

I will illustrate how one can view Euler's identity (and its generalizations) geometrically through enumerating integer points in certain polyhedra. This gives rise to “short” proofs of known theorems, as well as new identities.

This is joint work with Ben Braun (Kentucky).

seminars/comb/comb_abs_20140902.txt · Last modified: 2014/08/31 15:36 by qiao

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported