Abstract:
 The Springer numbers are defined in connection with the irreducible Coxeter groups. Some combinatorial interpretations of the Springer numbers of type A, B and D, have been found by Arnol'd in terms of snakes. In the first part of this talk, we introduce the relationship between some subclasses of snakes, lattice paths and polynomials associated with the successive derivatives of trigonometric functions.
According some particular purposes, one can define some inversion statistics, $inv^W(\pi)$, $inv^O(\pi)$, $inv^I(\pi)$ and the cycle number $cn(\pi)$ on a given snake $\pi$. Let the sign of a snake is the parity of the inversions or number of cycles, and the sign imbalance of a subclass of snakes be the sum of the sign of all the snakes in it. In the second part, we compute the sign imbalance of some subclasses of snakes with respect to $inv^W(\pi)$, $inv^O(\pi)$, $inv^I(\pi)$ and $cn(\pi)$.
{Key Words:} Springer number, snake, inversion, sign imbalance
{AMS Subject Classification:} 05A05, 05A19
