Springer Numbers and Arnold Families Revisited

TL;DR

This paper revisits Springer numbers and Arnold families, introducing polynomial arrays and new combinatorial objects that extend Arnold's original signed permutation concepts, with applications to derivatives of tangent and secant functions.

Contribution

It provides polynomial arrays for Springer number calculations, introduces new Arnold families, and connects these to derivatives of tangent and secant functions.

Findings

Derived polynomial arrays for Springer numbers.

Introduced new Arnold families of combinatorial objects.

Connected combinatorial structures to derivatives of trigonometric functions.

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 Andr\'{e} permutations and Simsun permutations.