The enumeration of permutations whose posets have a maximum element

Plane posets, special posets, and permutations

We study the self-dual Hopf algebra HSP of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from HSP to the Hopf algebra of free quasi-symmetric functions FQSym given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to FQSym; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomor...

Partitions, permutations and posets

1. Partitions For the definition of (number) partition, Ferrers diagram and Young tableaux, conjugate partition see Chapter 6 of R. Stanley’s Algebraic Combinatorics book. This section is based (more or less) on the treatment of the book A Course in Combinatorics by Van Lint and Wilson (Chapter 15). Proposition 1.1. Let pk(n) be the number of partitions of n into at most k parts. Let p(n) be th...

The Enumeration of Simple Permutations

The enumeration of simple permutations

A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coefficients of this series are not P -recursive, an asymptotic expansion for ...

Enumeration of Pin-Permutations

In this paper, we study the class of pin-permutations, that is to say of permutations having a pin representation. This class has been recently introduced in [16], where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of t...