ABSTRACT
We discuss rational, algebraic, and https://www.w3.org/1998/Math/MathML"> d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math0_8.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> -finite power series, and present examples for permutation classes whose generating functions are of those kinds. New to this edition is the result that the generating function counting the elements of length https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math0_9.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in a given principal permutations class is almost never rational. Significant new results in universal permutations, superpatterns, and their variations are included in the exercises. We also survey packing densities and frequency sequences. The latter are sequences https://www.w3.org/1998/Math/MathML"> a 0 , a 1 , ⋯ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math0_10.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where https://www.w3.org/1998/Math/MathML"> a i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math0_11.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the number of permutations of a given length that contain https://www.w3.org/1998/Math/MathML"> i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429274107/76d8f3df-fdac-4af3-ad29-e5f6ca3c54a5/content/math0_12.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> copies of a given pattern.
