Minimal and maximal plateau lengths in Motzkin paths

Helmut Prodinger (1) , Stephan Wagner (1)

1 DMS - Department of Mathematical Sciences [Matieland, Stellenbosch Uni.]

The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters maximal plateau length, horizontal segment and maximal horizontal segment as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

Mots clés: Motzkin paths; singularity analysis; Mellin transform; bootstrapping; unary-binary trees; Gumbel distribution

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Géométrie algorithmique [cs.CG]

Helmut Prodinger, Stephan Wagner. Minimal and maximal plateau lengths in Motzkin paths. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. pp.389-402, ⟨10.46298/dmtcs.3520⟩. ⟨hal-01184768⟩