The Littlewood-Richardson rule is a well-known procedure to calculate the product of two Schur functions, such as two elementary symmetric functions (see Macdonald (1995)).

b][X + [M/z]] we will use the additivity of the elementary symmetric functions.

Furthermore, the elementary symmetric functions are homogeneous so

q] for all strings of length at most m and over the first t elementary symmetric functions.

We will show that the indices of certain "critical" elementary symmetric functions determine the remaining elementary symmetric function values.

Our initial interest in the elementary symmetric functions stems from the counting of degree n monic irreducible polynomials over finite fields with prescribed coefficients for [x.

D]) consists in iterating their duplication theorems until their variables are nearly equal and then expanding in a series of elementary symmetric functions of the small differences between the variables.

Because the function is symmetric in the Z's, it can be expanded in elementary symmetric functions [E.

Their main examples of these Hopf-power chains were inverse shuffling (from the free associative algebra, with states indexed by its usual word basis) and rock-breaking (from the algebra of symmetric functions, with states indexed by the

elementary symmetric functions {[e.

Parkhurst and James [79] tabulate zonal polynomials of order 1 through 12 in terms of sums of powers and in terms of

elementary symmetric functions.

i] denote the complete homogeneous and

elementary symmetric functions of degree i respectively.

n] in terms of the

elementary symmetric functions [e.