{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:16:50Z","timestamp":1759940210470,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper we provide a bijective proof of a theorem of Garsia and Gessel describing the generating function of the major index over the set of all permutations of $[n]=\\{1,...,n\\}$ which are shuffles of given disjoint ordered sequences $\\pi_1,...,\\pi_k$ whose union is $[n]$.  The proof is based on a result (an \"insertion lemma\") of Haglund, Loehr, and Remmel which describes the change in major index resulting from the insertion of a given new element in any place in a given permutation.  Using this lemma we prove the theorem by establishing a bijection between shuffles of ordered sequences and a certain set of partitions.  A special case of Garsia and Gessel's theorem provides a proof of the equidistribution of major index and inversion number over inverse descent classes, a result first proved bijectively by Foata and Schutzenberger in 1978.  We provide, based on the method of our first proof, another bijective proof of this result.<\/jats:p>","DOI":"10.37236\/336","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:14:01Z","timestamp":1578716041000},"source":"Crossref","is-referenced-by-count":1,"title":["A Bijective Proof of a Major Index Theorem of Garsia and Gessel"],"prefix":"10.37236","volume":"17","author":[{"given":"Mordechai","family":"Novick","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,4,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r64\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r64\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:54:09Z","timestamp":1579305249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/336","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,4,19]]},"article-number":"R64"}}