Han's Bijection via Permutation Codes

We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $\pi$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $\sigma$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Bj\"orner and Wachs.Comment: 12 pages, to appear in European J. Combi

Trees, functional equations, and combinatorial Hopf algebras

One of the main virtues of trees is to represent formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in power series rings. When analyzed in terms of combinatorial Hopf algebras, the simplest examples yield interesting algebraic identities or enumerative results.Comment: 14 pages, LaTE

The Weak Bruhat Order and Separable Permutations

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, \pi]$ and $[\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets

An analogue of the plactic monoid for binary search trees

We introduce a monoid structure on a certain set of labelled binary trees, by a process similar to the construction of the plactic monoid. This leads to a new interpretation of the algebra of planar binary trees of Loday-Ronco.Comment: 4 pages, LaTex, Frenc