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Báo cáo toán học: "MacMahon-type Identities for Signed Even Permutations"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: MacMahon-type Identities for Signed Even Permutations. | MacMahon-type Identities for Signed Even Permutations Dan Bernstein Department of Mathematics The Weizmann Institute of Science Rehovot 76100 Israel dan.bernstein@weizmann.ac.il Submitted May 21 2004 Accepted Nov 15 2004 Published Nov 22 2004 Mathematics Subject Classifications 05A15 05A19 Abstract MacMahon s classic theorem states that the length and major index statistics are equidistributed on the symmetric group Sn. By defining natural analogues or generalizations of those statistics similar equidistribution results have been obtained for the alternating group An by Regev and Roichman for the hyperoctahedral group Bn by Adin Brenti and Roichman and for the group of even-signed permutations Dn by Biagioli. We prove analogues of MacMahon s equidistribution theorem for the group of signed even permutations and for its subgroup of even-signed even permutations. 1 Introduction A classic theorem by MacMahon 6 states that two permutation statistics namely the length or inversion number and the major index are equidistributed on the symmetric group Sn. Many rehnements and generalizations of this theorem are known today see 8 for a brief review . In 8 Regev and Roichman gave an analogue of MacMahon s theorem for the alternating group An c Sn and in 1 Adin Brenti and Roichman gave an analogue for the hyperoctahedral group Bn C2 o Sn. Both results involve natural generalizations of the Sn statistics having the equidistribution property. Our main result here Proposition 4.1 is an analogue of MacMahon s equidistribution theorem for the group of signed even permutations Ln C2 o An c Bn. Namely we dehne two statistics on Ln the L-length and the negative alternating reverse major index and show that they have the same generating function hence they are equidistributed. Our Main Lemma Lemma 4.6 shows that every element of Ln has a unique decomposition into a descent-free factor and a signless even factor. In 3 Biagioli proved an analogue of MacMahon s theorem for the group of .