  In this paper we apply a method of Robinson and Taulbee for computing
Kronecker coefficients together with other ingredients and show that the
multiplicity of each component in a Kronecker square can be obtained from an
evaluation of a certain polynomial, which depends only on the component and is
computed combinatorially. This polynomial has as many variables as the set of
isomorphism classes of connected skew diagrams of size at most the depth of the
component.
  We present two applications. The first is a contribution to Saxl conjecture,
which asserts that the Kronecker square of the staircase partition, contains
every irreducible character of the symmetric group as a component. We prove
that for any partition there is a piecewise polynomial function in one real
variable such that for all k, such that the multiplicity of this partition in
the Kronecker square of the staircase partition of size k is given by the
evaluation of the polynomial function in k. The second application is a proof
of a new stability property for Kronecker coefficients.
