论文标题
部分Lukasiewicz路径的列举
Enumeration of partial Lukasiewicz paths
论文作者
论文摘要
从$ \ bbb {n}^2 $开始，以$ x $ -Axis结束，由$ x $ -Axis结束，由集合$ \ {（1，k），k \ geq -1 \} $组成。我们给出这些路径的$ n $长度前缀（后缀）的生成函数和精确值，以高度$ k \ geq 0 $以给定类型的步骤结束。我们对最多$ t \ geq 0 $的高度前缀进行了类似的研究。使用有界高度路径的显式形式，我们渐近地评估平均高度。对于固定的$ k $和$ n \ to \ infty $，此数量的行为为$ \ sqrt {πn} $。最后，我们（以相同方式研究）替代Ukasiewicz路径的前缀，即确实包含两个连续的步骤，具有相同的方向。
Łukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length prefixes (resp. suffixes) of these paths ending at height $k\geq 0$ with a given type of step. We make a similar study for prefixes of height at most $t\geq 0$. Using the explicit forms for the paths of bounded height, we evaluate the average height asymptotically. For fixed $k$ and $n\to\infty$, this quantity behaves as $\sqrt{πn}$. Finally we study (in the same way) prefixes of alternate Łukasiewicz paths, i.e., Łukasiewicz paths that do contain two consecutive steps with the same direction.
