  The Gelfand-Tsetlin graph is an infinite graded graph that encodes branching
of irreducible characters of the unitary groups. The boundary of the
Gelfand-Tsetlin graph has at least three incarnations --- as a discrete
potential theory boundary, as the set of finite indecomposable characters of
the infinite-dimensional unitary group, and as the set of doubly infinite
totally positive sequences. An old deep result due to Albert Edrei and Dan
Voiculescu provides an explicit description of the boundary; it can be realized
as a region in an infinite-dimensional coordinate space.
  The paper contains a novel approach to the Edrei-Voiculescu theorem. It is
based on a new explicit formula for the number of semi-standard Young tableaux
of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). The
formula is obtained via the theory of symmetric functions, and new Schur-like
symmetric functions play a key role in the derivation.
