Some enumerations of non-trivial compositions of the differential operations and the directional derivative

TL;DR

This paper explores the enumeration of complex compositions of differential operations and directional derivatives in multi-dimensional space, providing recurrence relations for counting these higher-order non-trivial compositions.

Contribution

It introduces recurrence formulas for counting higher-order non-trivial compositions of differential operations and directional derivatives in  space.

Findings

Derived recurrence relations for composition counts

Extended enumeration to higher-order derivatives

Applicable to spaces with dimension n 

Abstract

This paper deals with the enumeration of the higher order non-trivial compositions of the differential operations and the directional derivative in the space $R^n$ ($n \geq 3$). We present the recurrences for a counting the higher order non-trivial compositions.