A geometric and combinatorial exploration of Hochschild lattices. (English) Zbl 1542.06006
Summary: Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by F. Chapoton [Algebr. Comb. 3, No. 2, 433–463 (2020; Zbl 1436.05122)]. A natural geometric realization of these lattices leads to some cell complexes introduced by S. Saneblidze [Topology Appl. 156, No. 5, 897–910 (2009; Zbl 1181.55009); “On the homology theory of the closed geodesic problem”, Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 25, 113–116 (2011)], called the Hochschild polytopes. We obtain several geometrical properties of the Hochschild lattices, namely we give cubic realizations, establish that these lattices are EL-shellable, and show that they are constructible by interval doubling. We also prove several combinatorial properties as the enumeration of their \(k\)-chains and compute their degree polynomials.
MSC:
| 06A07 | Combinatorics of partially ordered sets |
| 06B05 | Structure theory of lattices |
| 05A15 | Exact enumeration problems, generating functions |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Number of connected functions on n labeled nodes.Number of 1’s in all compositions of n+1.
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