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 \(\beta _n\)-snakes, and derived recurrence relations for enumerating the \(\beta _n\)-snakes starting with k. The results are presented in the form of double triangular arrays (\(v_{n,k}\)) of integers, \(1\le |k|\le n\). An Arnold family is a sequence of sets of such objects as \(\beta _n\)-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.