论文标题
三个象限步行的枚举，并在其他M骨骼锥上步行
Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones
论文作者
论文摘要
我们以小台阶局限于二维锥体（例如四分之一平面，四分之三的平面或缝隙平面）的列举。在四分之一平面的情况下，已经对未加权阶梯组的解决方案已经被充分了解，从某种意义上说，在哪种情况下，在哪些情况下，在哪种情况下，它是确切的，在哪些情况下是代数，D-finite或d-Elgebraic，并且在所有情况下都知道了确切的积分表达式。我们在更加一般的环境中获得了类似的结果：我们以任何正整数$ m $列举$ m $ $ m $的圆锥体，并从任何时候开始加权步骤。这项工作的主要突破是分析功能方程的推导，该方程表征了这些步道的生成函数，该方程类似于现在广泛用于四分之一平面步行的步道。在情况下，$ m = 3 $，与避免象限的步行相对应，我们为步行提供了精确的积分表达解决方案，并带有加权小步骤，这些步骤确定生成功能$ {\ sf c}（x，x，y; t）$计数这些步行。此外，对于步行的每个步骤集和起点，我们确定生成函数$ {\ sf c}（x，y; t）$是代数，d-finite还是D-Elgebraic作为$ x $和$ y $的函数。实际上，我们为任何$ M $ QUADRANT锥提供了这种类型的结果，这表明任何奇数$ M $都相同。对于$ m $，即使我们发现计算这些步道的生成功能在$ x $和$ y $中是D-Finite，并且仅当步行的起点与圆锥形的边界相同时，并且仅当步行的起点与轴的起点相同时。
We address the enumeration of walks with small steps confined to a two-dimensional cone, for example the quarter plane, three-quarter plane or the slit plane. In the quarter plane case, the solutions for unweighted step-sets are already well understood, in the sense that it is known precisely for which cases the generating function is algebraic, D-finite or D-algebraic, and exact integral expressions are known in all cases. We derive similar results in a much more general setting: we enumerate walks on an $M$-quadrant cone for any positive integer $M$, with weighted steps starting at any point. The main breakthrough in this work is the derivation of an analytic functional equation which characterises the generating function of these walks, which is analogous to one now used widely for quarter-plane walks. In the case $M=3$, which corresponds to walks avoiding a quadrant, we provide exact integral-expression solutions for walks with weighted small steps which determine the generating function ${\sf C}(x,y;t)$ counting these walks. Moreover, for each step-set and starting point of the walk we determine whether the generating function ${\sf C}(x,y;t)$ is algebraic, D-finite or D-algebraic as a function of $x$ and $y$. In fact we provide results of this type for any $M$-quadrant cone, showing that this nature is the same for any odd $M$. For $M$ even we find that the generating functions counting these walks are D-finite in $x$ and $y$, and algebraic if and only if the starting point of the walk is on the same axis as the boundaries of the cone.
