Complete Homogeneous Symmetric Polynomials

The $k$-th complete homogeneous symmetric polynomial in $m$ variables $h_{k,m}$ is the sum of all the monomials of degree $k$ in $m$ variables. They are related to the Symmetric powers of vector spaces. In this talk we will present some of their standard properties, some classic combinatorial results using the "stars and bars" argument, as well as an interesting result: the complete homogeneous symmetric polynomial applied to $(1+X_i)$ can be written as a linear combination of complete homogeneous symmetric poynomials in the $X_i$. To compute the coefficients of this linear combination, we extend the classic "stars and bars" argument.