Balanced Symmetric Functions over GF(p)GF(p)

Under mild conditions on $n,p$, we give a lower bound on the number of $n$-variable balanced symmetric polynomials over finite fields $GF(p)$, where $p$ is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that $X(2^t,2^{t+1}l-1)$ are the only nonlinear balanced elementary symmetric polynomials over GF(2), where $X(d,n)=\sum_{i_1<i_2<...<i_d}x_{i_1} x_{i_2}... x_{i_d}$, and we prove various results in support of this conjecture.Comment: 21 page

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 Sn corresponding to a balanced Young diagram λ (a Young diagram with at most View the MathML source rows and columns for some fixed constant C) in the limit as n tends to infinity