Parity statistics on restricted permutations and the Catalan--Schett polynomials

Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over $321$-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on $321$-avoiding permutations is solved and several combinatorial interpretations for the Catalan--Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.