Down-step statistics in generalized Dyck paths

07/30/2020

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by   Andrei Asinowski, et al.

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The number of down-steps between pairs of up-steps in k_t-Dyck paths, a recently introduced generalization of Dyck paths consisting of steps {(1, k), (1, -1)} such that the path stays (weakly) above the line y=-t, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between k_t-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.