Labeled Ballot Paths and the Springer Numbers

Original · EN

The Springer numbers are defined in connection with the irreducible root systems of type Bₙ, which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of Andre signed permutations, and by Arnol'd in terms of snakes of type Bₙ. We introduce the inversion code of a snake of type Bₙ and establish a bijection between labeled ballot paths of length n and snakes of type Bₙ. Moreover, we obtain the bivariate generating function for the number B(n,k) of labeled ballot paths starting at (0,0) and ending at (n,k). Using our bijection, we find a statistic α such that the number of snakes π of type Bₙ with α(π)=k equals B(n,k). We also show that our bijection specializes to a bijection between labeled Dyck paths of length 2n and alternating permutations on [2n].