Abstract
The set of permutations on a finite set can be given a lattice structure (known as the weak Bruhat order). The lattice structure is generalized to the set of words on a fixed alphabet \(\varSigma = \{\,x,y,z,\ldots \,\}\), where each letter has a fixed number of occurrences (these lattices are known as multinomial lattices and, in dimension 2, as lattices of lattice paths). By interpreting the letters \(x,y,z,\ldots \) as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, where \(d = \mathrm {card}(\varSigma )\).
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