It is well known that Riquier introduced the notion of multiplicative variables applied to order ideals to represent initial conditions of partial differential equations as series. On the other hands, Janet introduced the concept of involutive division. Following Riquier and Janet, we focus on the following problem.

Suppose one needs to compute a monomial ideal generated in some degree D and its escalier, knowing the Hilbert function and some monomials which must belong to the ideal/escalier. We give combinatorial tools to answer this question. First we define combinatorial decompositions of sets of terms, showing the criteria to decompose the set ${\mathcal{T}}_{\ge D}$ of the terms in degree greater or equal than D, into disjoint subsets called cones. This is done by assigning to each term t of degree D some multiplicative variables. The cone of t is then the set formed by t and all its multiples obtained multiplying it by all possible products of powers of multiplicative variables. Then, supposed one has selected a decomposition, we deal with the escalier/ideal partition problem, namely we study the rules which force a term to be in the ideal (or in the escalier) provided that another term does, so that both the ideal and the escalier turn out to be decomposed in disjoint cones by the decomposition set on the whole ${\mathcal{T}}_{\ge D}$.

众所周知，Riquier引入了应用于阶理想的乘法变量的概念，以将偏微分方程的初始条件表示为级数。另一方面，珍妮特介绍了渐进式除法的概念。继Riquier和Janet之后，我们关注以下问题。

假设一个人需要计算一个在D度及其escalier上生成的单项式理想，知道希尔伯特函数和一些必须属于该理想/ escalier的单项式。我们提供组合工具来回答这个问题。首先，我们定义项集合的组合分解，显示分解集合的条件${\mathcal{Ť}}_{\ge d}$程度大于或等于D的项分解为不相交的子集，称为圆锥。这是通过为度D的每个项t分配一些乘法变量来完成的。然后，t的圆锥就是由t构成的集合，并且将其所有倍数乘以所有可能的乘积变量幂的乘积。然后，假设一个人选择了分解，我们将处理escalier /理想划分问题，即研究规则，该条件强制一个术语处于理想状态（或escalier中），前提是另一个条件必须满足， escalier最终被分解集合分解为不相交的圆锥体${\mathcal{Ť}}_{\ge d}$。