Enumerative aspects of Caylerian polynomials

Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations, where a Cayley permutation is a word of positive integers such that if a number appears in the word then all positive integers less than that number also appear in the word. Using combinatorial species and sign-reversing involutions we derive counting formulas and generating functions for the Caylerian polynomials as well as for related refined polynomials.