We find a closed formula for the number of chains in Dyck lattices. Moreover, we determine a general formula to compute the number of saturated chains, and we apply it to find the number of saturated chains of length 2, 3 and 4. We also compute what we call the Hasse index (of order 2, 3 and 4) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2, 3 and 4) and the cardinality of the underlying poset.
Enumeration of chains and saturated chains in Dyck lattices / Ferrari, Luca; Emanuele, Munarini. - In: ADVANCES IN APPLIED MATHEMATICS. - ISSN 0196-8858. - STAMPA. - 62:(2015), pp. 118-140. [10.1016/j.aam.2014.09.003]
Enumeration of chains and saturated chains in Dyck lattices
FERRARI, LUCA;
2015
Abstract
We find a closed formula for the number of chains in Dyck lattices. Moreover, we determine a general formula to compute the number of saturated chains, and we apply it to find the number of saturated chains of length 2, 3 and 4. We also compute what we call the Hasse index (of order 2, 3 and 4) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2, 3 and 4) and the cardinality of the underlying poset.
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