2022/08/30 by Pham, Nhung
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.14496
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=\binomk2 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⋯, 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.