  The Bulgarian Solitaire rule induces a finite dynamical system on the set of
integer partitions of $n$. Brandt characterized and counted all cycles in its
recurrent set for any given $n$, with orbits parametrized by necklaces of black
and white beads. However, the transient behavior within each orbit has been
almost completely unknown. The only known case is when $n=\binom{k}{2}$ is a
triangular number, in which case there is only one orbit. Eriksson and Jonsson
gave an analysis for convergence of the structure as $k$ grows, and to what
extent the limit applied to the finite case. In this article, we generalize the
convergent structure for orbits of Bulgarian Solitaire system for any $n$. For
necklaces of the form $(BW)^k = BWBW\cdots$, we give the precise limit of the
generating functions as $k$ grows. For other necklaces, we prove that the
generating functions are rational and provide a bound for their denominator and
numerator degrees.
