关于保加利亚单人纸牌游戏及相关游戏的简短综述
A short survey the game Bulgarian solitaire and related games
AI总结：
本文对保加利亚单人纸牌游戏及相关游戏作简短综述，通过定义移位操作\(T\)描述游戏过程，介绍了\(T -\)循环划分，提及布兰特对其特征描述，还指出该游戏作为动力系统在\(N\)为三角形数时收敛到唯一不动点的特性。
AI中文摘要：
设\(N\)为任意正整数，\(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l)\)是\(N\)的一个长度为\(l\)的划分，即\(\sum_{i = 1}^l\lambda_i = N\)且\(\lambda_1 \geq \lambda_2 \ldots \lambda_l \geq 1\)。定义\(T(\lambda)\)为\(N\)的划分，其部分为\(l, \lambda_1 - 1, \lambda_2 - 1, \ldots, \lambda_l - 1\)（忽略可能出现的零）。从\(N\)的划分\(\lambda\)开始，通过反复应用移位操作\(T\)来描述保加利亚单人纸牌游戏，得到划分序列\(\lambda, T(\lambda), T^2(\lambda), \ldots\)。若对于某个\(i \geq 1\)有\(T(\mu) = \mu\)，则称\(N\)的划分\(\mu\)是\(T\)循环的。1982年布兰特对保加利亚单人纸牌游戏的所有\(T\)循环划分进行了特征描述。保加利亚单人纸牌游戏是正整数\(N\)的整数划分上的动力系统，当\(N = 1 + 2 + \cdots + k\)为三角形数时收敛到唯一不动点。本文对保加利亚单人纸牌游戏及其几种变体进行简短综述。
英文摘要：
Let $N$ be an arbitrary positive integer and let $λ=(λ_1, λ_2, \ldots, λ_l)$ be a partition of $N$ of length $l$, i.e., $\sum_{i=1}^lλ_i= N$ with parts $λ_1\ge λ_2\ldots λ_l\ge 1$. Define $T(λ)$ as the partition of $N$ with parts $l,λ_1-1λ_2-1,\ldots λ_l-1$,ignoring any zeros that might occur. Starting with a partition $λ$ of $N$, we describe Bulgarian solitaire by repeatedly applying the shift operation $T$ to obtain the sequence of partitions $$ λ, T(λ), T^2(λ),\ldots . $$ We say a partition $μ$ of $N$ is $T$-cyclic if $T(μ) = μ$ for some $i\ge 1$. In 1982 Brandt [9] characterized all $T$-cyclic partitions for Bulgarian solitaire. Bulgarian solitaire is a dynamical system on integer partition of a positive integer $N$ which converges to a unique fixed point if $N=1+2+\cdots +k$ is a triangular number. In this paper we present a short survey of the game Bulgarian solitaire and several variations of this game.
