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 (vn,k) of integers, 1 ≤ | k| ≤ n . An Arnold family is a sequence of sets of such objects as βn -snakes that are counted by (vn,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.
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