We prove a single sum formula for the linearization coefficients of the
Bessel polynomials. In two special cases we show that our formula reduces
indeed to Berg and Vignat's formulas in their proof of the positivity results
about these coefficients (Constructive Approximation, 27(2008), 15-32). As a
bonus we also obtain a generalization of an integral formula of Boros and Moll
(J. Comput. Appl. Math. 106 (1999), 361-368).

For a labeled tree on the vertex set $\set{1,2,...,n}$, define the local
direction of each edge $i-j$ as $i \to j$ if $i<j$. For a rooted tree there is
also a natural global direction of edges towards the root. The number of edges
pointing to a vertex is called its indegree. Thus the local (resp. global)
indegree sequence $\lambda = 1^{e_1}2^{e_2} ...$ of a tree on $\set{1,2,...,n}$
is a partition of $n-1$. We construct a bijection from (unrooted) trees to
rooted trees %which preserves the local indegree sequence.

We generalize two bijections due to Garsia and Gessel to compute the
generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$
and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a
symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$,
$\ides_G$, $\imaj_G$, and $\icol$ denote the number of descents, length, major
index, color weight, inverse descents, inverse major index, and inverse color
weight, respectively.

We give a new description of the flag major index, introduced by Adin and
Roichman, by using a major index defined by Reiner. This allows us to establish
a connection between an identity of Reiner and some more recent results due to
Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel
by computing the four-variate generating series of descents, major index,
length, and number of negative entries over Coxeter groups of type $B$ and $D$.