The authors connect k-coloured Motzkin paths to odd-height prefixes and supply a linear-time random generation algorithm.
Motzkin (1948): Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products
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Random Generation of $k$-coloured Motzkin Paths
The authors connect k-coloured Motzkin paths to odd-height prefixes and supply a linear-time random generation algorithm.