A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of a permutation pattern

In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrence of τ where τ starts with 1. In particular, we study the generating function ∑ n ≥ 0 t n n ! ∑ σ ∈ NM n ( 1324 … p ) x LRmin ( σ ) y 1 + des ( σ ) where p ≥ 4 , NM n ( 1324 … p ) is the set of permutations σ in the symmetric group S n which has no consecutive occurrences of 1324 … p , des ( σ ) is the number of descents of σ and LRmin ( σ ) is the number of left-to-right minima of σ . We show that for any p ≥ 4 , this generating function is of the form ( 1 U ( t , y ) ) x where U ( t , y ) = ∑ n ≥ 0 U n ( y ) t n n ! and the coefficients U n ( y ) satisfy some simple recursions depending on p . As an application of our results, we compute explicit generating functions for the number of permutations of S n that have no consecutive occurrences of the pattern 1324 … p and have exactly k descents for k = 1 , 2 .