Enumerating Permutations by their Run Structure

Understand

Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations.

• In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find that they can always be decomposed into so-called `atomic' permutations introduced in this work.
• This decomposition allows us to enumerate the (circular) permutations of a subset of the natural numbers by the length of their runs.
• Furthermore, we rederive, in an elementary way and using the methods developed here, a result due to Kitaev on the enumeration of valleys.