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Counting fundamental paths in certain Garside semigroups. (English) Zbl 1147.20049

Summary: For elements \(a,b\) of a monoid, define the word \(p_k(a,b)=abab\cdots\) 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 \(\langle a,b\mid p_k(a,b)=p_k(b,a)\rangle\). This number is related to counting certain paths in the \(\mathbb{N}\times\mathbb{N}\) 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 \(\Delta\), and similarly find the number of words equal in \(G\) to \(\Delta^n\).

MSC:

20M05 Free semigroups, generators and relations, word problems
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
05A15 Exact enumeration problems, generating functions
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Online Encyclopedia of Integer Sequences:

a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k).

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