We also show that the ordinary generating functions for j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions.

Most recently, Marberg (10) extended the result to coloured set partitions with a novel way of proving that the ordinary generating functions of j- noncrossing, k-nonnesting, r-coloured partitions according to size n are rational functions.

Since crossing and nesting statistics involves arcs, we define an r-coloured permutation parallel to (10) as a pair, (S, [phi]) consisting of a permutation of [n] and an arc-colour assigning map [phi] : Arc(S) [right arrow] [r], and use a capital Greek letter, [summation], to denote these objects.

Given an r-coloured permutation [summation] = (S, [phi]), let the set of openers (resp.

As customary in the literature, we let NC[N.sub.j,k](n, r) denote the number of all r-coloured, j-noncrossing, k-nonnesting permutations of [n].

However, another interpretation of Marberg's multigraphs [G.sub.j,k,r] in terms of the types of vertices and colours of edges leads to the multigraphs for r-coloured permutations which permits the application of transfer matrix method to draw the same conclusion: The ordinary generating function, [[summation].sub.n [greater than or equal to] 0] NC[N.sub.j,k] (n, r) [x.sup.n] for j-noncrossing, k-nonnesting, r-coloured permutations is rational.

Proof: We show an involution between the set of r-coloured permutations of [n] with maximal crossing number j, nesting number k and those with maximal crossing number k and nesting number j.

Given an r-coloured permutation of [n], say [summation] = (S, [phi]), first consider its corresponding uncoloured permutation S.

Finally, the combination of all r involutions, one for each colour, produces the desired r-coloured permutation such that for each colour, crossing number and nesting number are switched.

Before we enumerate r-coloured permutations, a quick overview of Marberg's approach for the enumeration of coloured set partitions helps set the stage for a new interpretation.

To construct [G.sub.2,2,2], we require four vertices: still [v.sub.0] as the initial state for no opener, but also two states indicating one r-coloured (r [member of] [2]) opener, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.4 Proof of Rationality through Multigraphs for r-coloured permutations