Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Dae San Kim 14 February 2006 2006 2 14 157 160 721-5123-Kim-2005 10.wxyz/C2005V72P1p157 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5123-Kim/ We evaluate in a simple and direct manner a multiple character sum, a special case of which can also be derived from the M\"{o}bius inversion and a result of Hanlon. 11L10,11L40,11T24 MR2162301 02212193 This work was supported by the Basic Research Program of the Korea Science and Engineering Foundation under Grant R01-2002-000-00083-0(2004). A.R. Calderbank, P. Hanlon and R.W. Robinson Proc. London Math. Soc. MR850222 A.R. Calderbank, P. Hanlon and R.W. Robinson; Partitions into even and odd block size and some unusual characters of the symmetric groups, \textit{Proc. London Math. Soc.} \textbf{53} (1986), pp.~288--320. P. Hanlon Pacific J. Math. MR637975 P. Hanlon; The fixed-point partition lattices, \textit{Pacific J. Math.} \textbf{96} (1981), pp.~319--341. P. Hanlon J. Algebra MR769584 P. Hanlon; The characters of the wreath product group acting on the homology groups of the Dowling lattices, \textit{J. Algebra} \textbf{91} (1984), pp.~430--463. P. Hanlon Trans. Amer. Math. Soc. MR1014249 P. Hanlon; The generalized Dowling lattices, \textit{Trans. Amer. Math. Soc.} \textbf{325} (1991), pp.~1--37.