Adjacencies in Words

Based Based on on two two inversion inversion formulas formulas for for enumerating enumerating words words in in the the free free monoid monoid by by adjacencies, adjacencies, we we present present a a new new approach approach to to a a class class of of permutation permutation problems problems having having Eulerian-type Eulerian-type generating generating functions. functions. We We also also show show that that a a specialization specialization of of one one of of the the inversion inversion formulas formulas gives gives Diekert's Diekert's lifting lifting to to the the free free monoid monoid of of an an inversion inversion theorem theorem due due to to Cartier Cartier and and Foata. Foata. There There are are a a number number of of powerful powerful theories theories of of inversion inversion [9, [9, 10, 10, 13, 13, 16) 16) for for dealing dealing with with combinatorial combinatorial objects objects having having generating generating functions functions of of Eulerian Eulerian-type-type Using Using two two such such inversion inversion formulas, formulas, we we present present new new derivations derivations of of Stanley's Stanley's [13] [13] generating generating functions functions for for generalized generalized q-Eulerian q-Eulerian and and q-Euler q-Euler polyno­ polyno­ mials mials on on r-tuples r-tuples of of permutations. permutations. We We further further indicate indicate how how one one of of the the inversion inversion formulas formulas gives gives Diekert's Diekert's [5] [5] lifting lifting to to the the free free monoid monoid of of an an