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For any pattern α of length at most two, we enumerate equivalence classes of Lukasiewicz paths of length n ≥ 0 where two paths are equivalent whenever the occurrence positions of α are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Lukasiewicz paths.

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Abstract. In a recent paper [2], the authors provide enumerating results for equivalence classes of permutations modulo excedances. In this paper we investigate two other equivalence relations based on descents and left-to-right maxima.Enumerating results are presented for permutations, involutions, derangements, cycles and permutations avoiding one pattern of length three.Mathematics Subject Classication(2010). 05A05.