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Path counting and random matrix theory. (English) Zbl 1031.05017
Electron. J. Comb. 10, Research paper R43, 16 p. (2003); printed version J. Comb. 10, No. 4 (2003).
Summary: 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 \(\beta\)-Hermite and \(\beta\)-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.

MSC:
05A19 Combinatorial identities, bijective combinatorics
15B52 Random matrices (algebraic aspects)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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