Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach.

翻译：根树对于通过所谓的B级数描述数值格式至关重要。它们也广泛用于粗糙分析中以展开奇异随机偏微分方程（SPDEs）的解。当考虑标量值方程时，最高效的组合集是多重指标。本文研究了是否存在介于多重指标和根树之间的中间组合集。我们给出了一个否定结果：在维度$d\neq 1$的情况下，除了根树本身外，不存在能够编码基本微分、且与根树及多重指标兼容的组合集。这并未终结关于此类组合集存在性的讨论，但表明无法通过一种朴素而自然的方法获得它。