Generating functions for descents over permutations which avoid sets of consecutive patterns

We extend the reciprocity method of Jones and Remmel ([Discrete Math. 313 (2013), 2712–2729] and [Pure Math. Appl. 24 (2013), 151–178]) to study generating functions of the form ∑ n≥0 t n! ∑ σ∈NMn(Γ) xy where Γ is a set of permutations which start with 1 and have at most one descent, NMn(Γ) is the set of permutations σ in the symmetric group Sn which have no Γ-matches, 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 this generating function is of the form ( 1 UΓ(t,y) )x where UΓ(t, y) = ∑ n≥0 UΓ,n(y) tn n! and the coefficients UΓ,n(y) satisfy some simple recursions in the case where Γ equals {1324, 123}, {1324 · · ·p, 12 · · · (p − 1)} and p ≥ 5, or Γ is the set of permutations σ = σ1 · · ·σn of length n = k1 + k2 where k1, k2 ≥ 2, σ1 = 1, σk1+1 = 2, and des(σ) = 1.