A Bijective Proof of Stanley's Shuffling Theorem

For two permutations o and u on disjoint sets of integers, consider forming a permutation on the combined sets by "shuffling" o and u (i.e., a and co appear as subsequences). Stanley [10], by considering P-partitions and a g-analogue of Saalschutz's 3F2 summation, obtained the generating function for shuffles of o and u with a given number of falls (an element larger than its successor) with respect to greater index (sum of positions of falls). It is a product of two ^-binomial coefficients and depends only on remarkably simple parameters, namely the lengths, numbers of falls and greater indexes of o and u. A combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce a and u to canonical permutations, for which a direct evaluation of the generating function is given.