《Chow and Mansour---A Carlitz identity for the wreath product Cr Sn》.pdf

Advances in Applied Mathematics 47 (2011) 199–215 Contents lists available at ScienceDirect Advances in Applied Mathematics /locate/yaama A Carlitz identity for the wreath product Cr Sn Chak-On Chow a , Toufik Mansour b,∗ a Department of Mathematics and Information Technology, Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong b Department of Mathematics, University of Haifa, 31905 Haifa, Israel a r t i c l e i n f o a b s t r a c t Article history: We present in this work a new flag major index fmajr for Received 6 November 2009 the wreath product G = C S , where C is the cyclic group r ,n r n r Accepted 3 June 2010 of order r and S is the symmetric group on n letters. We n Available online 10 August 2010 prove that fmajr is equidistributed with the length function on G and that the generating function of the pair (des , fmaj ) MSC: r ,n r r primary 05A15 over Gr ,n , where desr is the usual descent number on Gr ,n , satisfies secondary 05A19, 05A05, 05E05, 05E10 a “natural” Carlitz identity, thus unifying and generalizing earlier