论文标题
限制保加利亚纸牌轨道生长的行为
Limiting behavior in growth of Bulgarian Solitaire orbits
论文作者
论文摘要
保加利亚纸牌规则在$ n $的整数分区集中诱导有限的动力系统。勃兰特（Brandt）对任何给定的$ n $进行了反复设置的所有周期的表征，并用黑色和白色珠子项链进行了参数。但是，每个轨道内的瞬态行为几乎是完全未知的。唯一已知的情况是当$ n = \ binom {k} {2} $是三角形数字时，在这种情况下只有一个轨道。埃里克森（Eriksson）和琼森（Jonsson）分析了该结构的融合，随着$ k $的增长，以及在有限案例中应用的极限在多大程度上。在本文中，我们将保加利亚纸牌系统轨道的收敛结构概括为任何$ n $。对于表格$（BW）^K = BWBW \ CDOTS $的项链，我们将生成功能的确切限制为$ K $成长。对于其他项链，我们证明生成函数是合理的，并为其分母和分子程度提供了界限。
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.
