Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.6

Skew Dyck Paths Having no Peaks at Level 1

Helmut Prodinger
Stellenbosch University
Department of Mathematical Sciences
7602 Stellenbosch
South Africa
(National Institute for Theoretical and Computational Sciences)
South Africa


Skew Dyck paths are a variation of Dyck paths, where in addition to the steps (1, 1) and (1, –1), a south-west step (–1, –1) is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east step, we end up with decorated Dyck paths. Sequence A128723 of the On-Line Encyclopedia of Integer Sequences (OEIS) considers such paths where peaks at level 1 are forbidden. We provide a thorough analysis of a more general scenario, namely partial decorated Dyck paths, ending on a prescribed level j, both from left-to-right and from right-to-left (decorated Dyck paths are not symmetric). The approach is completely based on generating functions.

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(Concerned with sequences A128723.)

Received January 3 2022; revised version received January 17 2022. Published in Journal of Integer Sequences, January 26 2022.

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