Symmetric unimodal expansions of excedances in colored permutations

We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $\gamma$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the literature.Comment: 27 pages, 10 figure

On the enumeration of rooted trees with fixed size of maximal decreasing trees

Let \T_{n} be the set of rooted labeled trees on $\set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement \T_{n,k} of \T_n, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.Comment: 10 pages, 1 figure

A Generalized Enumeration of Labeled Trees and Reverse Pr\"ufer Algorithm

A {\em leader} of a tree $T$ on $[n]$ is a vertex which has no smaller descendants in $T$. Gessel and Seo showed \sum_{T \in \mathcal{T}_n}u^\text{(# of leaders in $T$)} c^\text{(degree of 1 in $T$)}=u P_{n-1}(1,u,cu), which is a generalization of Cayley formula, where $\mathcal{T}_n$ is the set of trees on $[n]$ and $$P_n(a,b,c)=c\prod_{i=1}^{n-1}(ia+(n-i)b+c).$$ Using a variation of Pr\"ufer code which is called a {\em RP-code}, we give a simple bijective proof of Gessel and Seo's formula.Comment: 5 pages, 3 figure