Noncommutative crossing partitions

We define and study noncommutative crossing partitions which are a generalization of non-crossing partitions. By introducing a new cover relation on binary trees, we show that the partially ordered set of noncommutative crossing partitions is a graded lattice. This new lattice contains the Kreweras lattice, the lattice of non-crossing partitions, as a sublattice. We calculate the M\"obius function, the number of maximal chains and the number of $k$-chains in this new lattice by constructing an explicit $EL$-labeling on the lattice. By use of the $EL$-labeling, we recover the classical results on the Kreweras lattice. We characterize two endomorphism on the Kreweras lattice, the Kreweras complement map and the involution defined by Simion and Ullman, in terms of the maps on the noncommutative crossing partitions. We also establish relations among three combinatorial objects: labeled $k+1$-ary trees, $k$-chains in the lattice, and $k$-Dyck tilings.Comment: 45 page

Heaps of pieces for lattice paths

We study heaps of pieces for lattice paths, which give a combinatorial visualization of lattice paths. We introduce two types of heaps: type $I$ and type $II$. A heap of type $I$ is characterized by peaks of a lattice path. We have a duality between a lattice path $\mu$ and its dual $\overline{\mu}$ on heaps of type $I$. A heap of type $II$ for $\mu$ is characterized by the skew shape between the lowest path and $\mu$. We give a determinant expression for the generating function of heaps for general lattice paths, and an explicit formula for rational $(1,k)$-Dyck paths by using the inversion lemma. We introduce and study heaps in $k+1$-dimensions which are bijective to heaps of type $II$ for $(1,k)$-Dyck paths. Further, we show a bijective correspondence between type $I$ and type $II$ in the case of rational $(1,k)$-Dyck paths. As another application of heaps, we give two explicit formulae for the generating function of heaps for symmetric Dyck paths in terms of statistics on Dyck paths and on symmetric Dyck paths respectively.Comment: 31 page