  Laguerre histories (restricted or not) are certain weighted Motzkin paths
with two types of level steps. They are, on one hand, in natural bijection with
the set of permutations, and on the other hand, yield combinatorial
interpretations for the moments of Laguerre polynomials via Flajolet's
combinatorial theory of continued fractions. In this paper, we first introduce
a reflection-like involution on restricted Laguerre histories. Then, we
demonstrate its power by composing this involution with three bijections due to
Fran\ccon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of
equidistribution results involving various (multiset-valued) permutation
statistics follow from these applications. As byproducts, seven apparently new
Mahonian statistics present themselves; new interpretations of known Mahonian
statistics are discovered as well. Finally, in our effort to show the
interconnections between these Mahonian statistics, we are naturally led to a
new link between the variant Yan-Zhou-Lin bijection and the Kreweras
complement.
