id: 06015025
dt: j
an: 06015025
au: Chen, William Y.C.; Fan, Neil J.Y.; Jia, Jeffrey Y.T.
ti: Labeled ballot paths and the Springer numbers.
so: SIAM J. Discrete Math. 25, No. 4, 1530-1546 (2011).
py: 2011
pu: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
la: EN
cc: 
ut: Springer number; snake of type $B_n$; labeled ballot path; labeled Dyck
    path; bijection
ci: 
li: doi:10.1137/100808460
ab: Summary: The Springer numbers are defined in connection with the
    irreducible root system of type $B_n$ and 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 André 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 $α$ such that the number of snakes $π$ of type $B_n$ 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 $[2n]$.
rv: