Springer Numbers and Arnold Families Revisited

Abstract

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.