Springer Numbers and Arnold Families Revisited

Abstract

For the calculation of Springer numbers (of root systems) of type Bn and Dn, Arnold introduced a signed analogue of alternating permutations, called βn-snakes, and derived recurrence relations for enumerating the βn-snakes starting with k. The results are presented in the form of double triangular arrays (v n,k ) of integers, 1 ≤ |k| ≤ n. An Arnold family is a sequence of sets of such objects as βn-snakes that are counted by (v n,k ). As a refinement of Arnold's result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of tan x and sec x, established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.