Limiting behavior in growth of Bulgarian Solitaire orbits

TL;DR

This paper investigates the limiting behavior of Bulgarian Solitaire orbits, generalizing previous results to all cases and analyzing the structure of generating functions for different necklace configurations.

Contribution

It extends the understanding of Bulgarian Solitaire orbits beyond the special case of triangular numbers, providing limits and bounds for generating functions in general.

Findings

Limit of generating functions for necklaces of the form (BW)^k is precisely determined.

Generating functions for other necklaces are rational with bounded degrees.

The analysis applies to all orbit types, not just special cases.

Abstract

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.