[Paper Review] Enumeration of edges in some lattices of paths

Luca Ferrari, Emanuele Munarini|arXiv (Cornell University)|Mar 30, 2012

Advanced Combinatorial Mathematics9 references4 citations

TL;DR

This paper enumerates the number of edges in the Hasse diagrams of various lattice path posets, including Dyck, Motzkin, Schröder, and Young lattices, using path decomposition and generating functions. It derives a general formula for the number of edges in any Young lattice $\mathcal{Y}_\lambda$ via corner cell contributions, proving that the Hasse index of classical path lattices is asymptotically equivalent to that of Boolean lattices.

ABSTRACT

We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder and Grand Schröder lattices. Finally, we give a general formula for the number of edges in an arbitrary Young lattice (which can be interpreted in a natural way as a lattice of paths).

Motivation & Objective

• To enumerate the number of edges in the Hasse diagrams of distributive lattices arising from classical lattice paths such as Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder, and Grand Schröder paths.
• To derive a general closed-form formula for the number of edges in the Hasse diagram of an arbitrary Young lattice $\mathcal{Y}_\lambda$ associated with a partition $\lambda$.
• To investigate the asymptotic behavior of the Hasse index (edge-to-vertex ratio) in these lattices and compare it to that of Boolean lattices.
• To establish connections between path decomposition techniques and the enumeration of saturated chains in Dyck lattices through generating series.

Proposed method

• The authors use path decomposition techniques to derive generating series for edge counts in classical path lattices, leveraging known enumerative sequences such as Catalan and central binomial coefficients.
• For Young lattices, they define corner cells and decompose the lattice based on the partitions $\lambda_{ij}^{\swarrow}$ and $\lambda_{ij}^{\nearrow}$ associated with each cell $ (i,j) $ in the Ferrers diagram.
• They derive the key formula $ \ell(\mathcal{Y}_\lambda) = \sum_{(i,j)\in\Phi_\lambda} |\mathcal{Y}_{\lambda_{ij}^{\swarrow}}| \cdot |\mathcal{Y}_{\lambda_{ij}^{\nearrow}}| $, which counts edges by summing over all corner cells and their associated sublattices.
• The generating series for the edge count in $ L(m,n) = \mathcal{Y}_{(n,\dots,n)} $ is computed as $ \frac{xy}{(1-x-y)^2} $, leading to the closed-form expression $ \ell(L(m,n)) = \binom{m+n-1}{n}n $.
• They analyze the Hasse index (edge-to-vertex ratio) and show that for all classical path lattices, it is asymptotically equivalent to that of Boolean lattices.

Experimental results

Research questions

• RQ1 What is the exact number of edges in the Hasse diagram of the Dyck lattice $\mathcal{D}_n$?
• RQ2 How can the number of edges in the Hasse diagram of an arbitrary Young lattice $\mathcal{Y}_\lambda$ be computed for any partition $\lambda$?
• RQ3 What is the asymptotic behavior of the Hasse index (edge-to-vertex ratio) in classical path lattices such as Dyck and Motzkin lattices?
• RQ4 How does the Hasse index of path lattices compare to that of Boolean lattices?

Key findings

• The number of edges in the Hasse diagram of the Young lattice $ L(m,n) $ is given by $ \ell(L(m,n)) = \binom{m+n-1}{n}n $, which is derived from the generating series $ \frac{xy}{(1-x-y)^2} $.
• For any partition $ \lambda $, the number of edges in $ \mathcal{Y}_\lambda $ is $ \sum_{(i,j)\in\Phi_\lambda} |\mathcal{Y}_{\lambda_{ij}^{\swarrow}}| \cdot |\mathcal{Y}_{\lambda_{ij}^{\nearrow}}| $, where the sum is over all cells in the Ferrers diagram of $ \lambda $.
• The Hasse index of the Dyck lattice $ \mathcal{D}_n $ is asymptotically equivalent to that of the Boolean lattice of rank $ n $, and the same holds for Grand Dyck, Motzkin, Grand Motzkin, Schröder, and Grand Schröder lattices.
• The Hasse index of all classical path lattices is either equal, asymptotically equivalent, or asymptotically quasi-equivalent to that of Boolean lattices.