Labeled Ballot Paths and the Springer Numbers

TL;DR

This paper establishes a bijection between labeled ballot paths and snakes of type B_n, providing new combinatorial interpretations and generating functions for Springer numbers and related structures.

Contribution

It introduces the inversion code of type B_n snakes and links labeled ballot paths with these snakes, offering new combinatorial insights and a specialized bijection.

Findings

Derived a bivariate generating function for B(n,k)

Established a bijection between labeled ballot paths and snakes of type B_n

Connected labeled Dyck paths with alternating permutations

Abstract

The Springer numbers are defined in connection with the irreducible root systems of type $B_n$, 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 $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length n and snakes of type $B_n$. 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 $\alpha$ such that the number of snakes $\pi$ of type $B_n$ with $\alpha(\pi)=k$ 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].