A Heyting Algebra on Dyck Paths of Type AA and BB

In this article we investigate the lattices of Dyck paths of type $A$ and $B$ under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the relative pseudocomplements using the lattice-theoretic definition is quite tedious. We give a combinatorial description of the Heyting algebra operations join, meet, and relative pseudocomplement in terms of height sequences, and we use these results to derive formulas for pseudocomplements and to characterize the regular elements in these lattices.Comment: Final version. 21 pages, 5 figure

A lattice on Dyck paths close to the Tamari lattice

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice

The Rank Enumeration of Certain Parabolic Non-Crossing Partitions

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton's $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.Comment: 31 pages, 5 figures. Comments are welcom

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds