Limiting Behavior in Growth of Bulgarian Solitaire Orbits

    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 coincides with the finite case for each $k$. 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.