Counting Upper Interactions in Dyck Paths

A Dyck word w is a word over the alphabet {x, x} that contains as many letters x as letters x and such that any prefix contains at least as many letters x as letters x. The size of w is the number of letters x in w. A Dyck path is a walk in the plane, that starts from the origin, is made up of rises, i.e. steps (1, 1), and falls, i.e. steps (1,−1), remains above the horizontal axis and finishes on it. Figure 1 gives an example of a Dyck path of size 12. The Dyck path related to a Dyck word w is the walk obtained by representing a letter x by a rise, and a letter x by a fall. In this paper we identify the two notions. An upper interaction, respectively a lower interaction, in a Dyck word w is an occurrence of a factor xx, respectively xx, for any k ≥ 1. The example of Figure 1 contains 7 upper interactions and 9 lower interactions.