Upper bound on the characters of the symmetric groups for balanced Young   diagrams and a generalized Frobenius formula
  We study asymptotics of an irreducible representation of the symmetric group S_n corresponding to a balanced Young diagram \lambda (a Young diagram with at most C\sqrt{n} rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that |\chi^{\lambda}(\pi)| = | Tr \rho^{\lambda}(\pi)/Tr \rho^{\lambda}(e) | < [ D max(1,|\pi|^2/n) / \sqrt{n}} ]^{|\pi|}, where |\pi| denotes the length of a permutation (the minimal number of factors necessary to write \pi as a product of transpositions). Our main tool is an analogue of Frobenius character formula which holds true not only for cycles but for arbitrary permutations. 
Amarpreet Rattan, Piotr Sniady