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In this paper we provide a bijective proof of a theorem of Garsia and Gesseldescribing the generating function of the major index over the set of allpermutations of [n]={1,...,n} which are shuffles of given disjoint orderedsequences whose union is [n]. Two special cases are singled out: If the singleelement j is inserted into any permutation P of the remaining elements of [n],then the theorem states that inserting j into P increases the major index of Pby some element of {0,1,...,n-1}, the increase determined uniquely by the indexof insertion. We provide a direct proof of this fact using an algorithm whichcalculates the increase at each index; this in turn leads to a bijective proofof MacMahon's 1916 result on the equidistribution of major index and inversionnumber over S_n. Using this special case we prove the general case of thetheorem by establishing a bijection between shuffles of ordered sequences and acertain set of partitions. In the second special case of interest, Garsia andGessel's theorem provides a proof of the equidistribution of major index andinversion number over inverse descent classes, a result first provedbijectively by Foata and Schutzenberger in 1978. We provide, based on themethod of our first proof, another bijective proof of this result.

A Bijective Proof of a Major Index Theorem of Garsia and Gessel