Narrow Results

For elements a, b of a monoid, define the word p_k(a,b) = abab⋯ of length k. We find the number of words in a, b which are equal to p_k(a,b)ⁿ in the Artin semigroup < a,b|p_k(a,b) = p_k(b,a) >. This number is related to counting certain paths in the ℕ × ℕ lattice. These Artin groups are examples of two generator Garside groups. We also define other examples of Garside groups G on more than two generators, having fundamental word Δ, and similarly find the number of words equal in G to Δⁿ.

We solve the problem of effectively computing the a-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type.