Báo cáo khoa học: Path counting and random matrix theory
We establish three identities involving Dyck paths and alternating Motzkin
paths, whose proofs are based on variants of the same bijection. We interpret
these identities in terms of closed random walks on the halfline. We explain how
these identities arise from combinatorial interpretations of certain properties of the
-Hermite and -Laguerre ensembles of random matrix theory. We conclude by
presenting two other identities obtained in the same way, for which finding combinatorial
proofs is an open problem....