The Enumeration of Lattice Paths With Respect to Their Number of Turns

Abstract

We survey old and new results on the enumeration of lattice paths in the plane with a given number of turns, including the recent developments on the enumeration of nonintersecting lattice paths with a given number of turns. Motivations to consider such enumeration problems come from various fields, e.g. probability, statistics, combinatorics, and commutative algebra. We show that the appropriate tool for treating turn enumeration of lattice paths is the encoding of lattice paths in terms of two-rowed arrays.Keywords and phrasesTurnslattice pathsnonintersecting lattice pathscoin tossingrun statisticsnon-crossing two-rowed arraysdeterminantal ringspfaffian ringsHilbert seriestableauxplane partitions