The q-exponential generating function for permutations by consecutive patterns and inversions

The inverse of Fedouʹs insertion-shift bijection is used to deduce a general form for the q-exponential generating function for permutations by consecutive patterns (overlaps allowed) and inversion number from a result due to Jackson and Goulden for enumerating words by distinguished factors. Explicit q-exponential generating functions are then derived for permutations by the consecutive patterns 12 … m , 12 … ( m − 2 ) m ( m − 1 ) , 1 m ( m − 1 ) … 2 , and by the pair of consecutive patterns ( 123 , 132 ) .