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The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating...
Motivated by the relation N m ( C n ) = ( m n + 1 ) N m ( A n − 1 ) , holding for the m -generalized Catalan numbers of type A and C , the connection between dominant regions of the m -Shi arrangement of type A n − 1 and C n is investigated. More precisely, it is explicitly shown how m n + 1 copies of the set of dominant regions of the m -Shi...
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