Determinantal representations of enumerative polynomials

Based on a determinantal formula for the higher derivative of a quotient of two functions, we first present the determinantal expressions of Eulerian polynomials and André polynomials. In particular, we discover that the Euler number (number of alternating permutations) can be expressed as a lower Hessenberg determinant. We then investigate the determinantal representations of the up-down run polynomials and the types $A$ and $B$ alternating run polynomials. As applications, we deduce several new recurrence relations. And then, we provide two determinantal representations for the alternating run polynomials of dual Stirling permutations. In particular, we discover a close connection between the alternating run polynomials of dual Stirling permutations and the type $B$ Eulerian polynomials.