## ABSTRACT

We count permutations with a given descent set, and permutations with a given number of descents. We prove an explicit formula for the Eulerian numbers, and show that the Eulerian polynomials have real roots only. As counting descents is equivalant to counting increasing runs, we then we turn to alternating runs and alternating subsequences of permutations, and prove analogous results for them. New to this edition is a proof that uses a group action to show that about half of the roots of the generating polynomial counting permutations of a given length with respect to the number of their alternating runs is equal to https://www.w3.org/1998/Math/MathML"> − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math00_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .