Springer Numbers and Arnold Families Revisited

For the calculation of Springer numbers (of root systems) of type B_n and D_n , Arnold introduced a signed analogue of alternating permutations, called β_n -snakes, and derived recurrence relations for enumerating the β_n -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 β_n -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.