\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{recrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \newtheorem{theo}{Theorem}[section] \newtheorem{lemma}[theo]{Lemma} \def\dire{dir{\hspace{-0.11em}\raisebox{-0.65ex}{\mbox{\scriptsize $\vec{e}$}}}\,} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm {\LARGE\bf The Compositions of Differential~Operations\\ \vskip .1in and the Gateaux~Directional~Derivative} \vskip 1cm \large Branko J. Male\v sevi\' c\footnote{Supported in part by the project MNTRS, Grant No. ON144020.} and Ivana V. Jovovi\'c\footnote{Ph.\ D. student, Faculty of Mathematics, University of Belgrade, Serbia.} \\ University of Belgrade \\ Faculty of Electrical Engineering\\ Bulevar kralja Aleksandra 73\\ Belgrade\\ Serbia \\ \href{mailto:malesh@EUnet.yu}{\tt malesh@EUnet.yu}\\ \href{mailto:ivana121@EUnet.yu}{\tt ivana121@EUnet.yu}\\ \end{center} \vskip .2in \begin{abstract} This paper deals with the number of meaningful compositions of higher order of differential operations and the Gateaux directional derivative. \end{abstract} \section{The compositions of differential operations of the space $\mathbb R^{\mbox{\footnotesize \textbf\textit{3}}}$} In the real three-dimensional space $\mathbb R^{3}$ we consider the following sets$:$ \begin{equation} \mbox{\rm A}_{0} = \{ f\!:\! \mathbb R^{3} \!\longrightarrow\! \mathbb R \, | \, f\!\in\!C^{\infty}(\mathbb R^{3}) \} \;\;\; \mbox{and} \;\;\; \mbox{\rm A}_{1} = \{ \vec{f}\!:\! \mathbb R^{3} \!\longrightarrow\! \mathbb R^{3} \, | \, \vec{f}\!\in\!\vec{C}^{\infty}(\mathbb R^{3}) \}. \end{equation} It is customary in vector analysis to consider $m=3$ basic differential operations on $\mbox{\rm A}_{0}$ and $\mbox{\rm A}_{1}$ \cite{Ivanov02}, namely$:$ \begin{equation} \begin{array}{l} \mbox{ \small $ \mbox{\normalsize \rm grad} \, \mbox{\normalsize $f$} = \mbox{\normalsize $\nabla_1$} \, \mbox{\normalsize $f$} \!=\! \left( \displaystyle\frac{\partial f}{\partial x_1} \;,\; \displaystyle\frac{\partial f}{\partial x_2} \;,\; \displaystyle \frac{\partial f}{\partial x_3} \right) : \mbox{\normalsize \rm A}_{0} \longrightarrow \mbox{\normalsize \rm A}_{1}$}, \\[2.5 ex] \mbox{ \small $ \mbox{\normalsize \rm curl} \, \vec{\mbox{\normalsize $f$}} = \mbox{\normalsize $\nabla_2$} \, \vec{\mbox{\normalsize $f$}} = \left( \displaystyle \frac{\partial f_3}{\partial x_2} \!-\! \displaystyle \frac{\partial f_2}{\partial x_3} \;,\; \displaystyle \frac{\partial f_1}{\partial x_3} \!-\! \displaystyle \frac{\partial f_3}{\partial x_1} \;,\; \displaystyle \frac{\partial f_2}{\partial x_1} \!-\! \displaystyle \frac{\partial f_1}{\partial x_2} \right) : \mbox{\normalsize \rm A}_{1} \longrightarrow \mbox{\normalsize \rm A}_{1}$}, \\[2.5 ex] \mbox{ \small $ \mbox{\normalsize \rm div} \, \vec{\mbox{\normalsize $f$}} = \mbox{\normalsize $\nabla_3$} \, \vec{\mbox{\normalsize $f$}} = \displaystyle \frac{\partial f_1}{\partial x_1}+ \displaystyle \frac{\partial f_2}{\partial x_2}+ \displaystyle \frac{\partial f_3}{\partial x_3}: \mbox{\normalsize \rm A}_{1} \longrightarrow \mbox{\normalsize \rm A}_{0} $ }. \end{array} \end{equation} Let us present the number of meaningful compositions of higher order over the set ${\cal A}_{3} = \{ \nabla_1, \nabla_2, \nabla_3\}$. It is familiar fact that there are $m=5$ compositions of the second order \cite[p.$\,$161]{Korn00}$:$ \begin{equation} \begin{array}{l} \Delta f = \mbox{div\,grad} \, f = \nabla_3 \circ \nabla_1\, f, \\[1.5 ex] \mbox{curl\,curl} \, \vec f = \nabla_2 \circ \nabla_2\, \vec f, \\[1.5 ex] \mbox{grad\,div} \, \vec f = \nabla_1 \circ \nabla_3\, \vec f, \\[1.5 ex] \mbox{curl\,grad} \, f = \nabla_2 \circ \nabla_1\, f = \vec 0, \\[1.5 ex] \mbox{div\,curl} \, \vec f = \nabla_3 \circ \nabla_2\, \vec f = 0. \end{array} \end{equation} Male\v sevi\' c \cite{HiOrd96} proved that there are $m=8$ compositions of the third order$:$ \begin{equation} \begin{array}{l} \mbox{grad\,div\,grad}\, f = \nabla_1 \circ \nabla_3 \circ \nabla_1\,f, \\[1.5 ex] \mbox{curl\,curl\,curl}\, \vec f = \nabla_2 \circ \nabla_2 \circ \nabla_2\,\vec f, \\[1.5 ex] \mbox{div\,grad\,div}\, \vec f = \nabla_3 \circ \nabla_1 \circ \nabla_3\,\vec f, \\[1.5 ex] \mbox{curl\,curl\,grad}\, f = \nabla_2 \circ \nabla_2 \circ \nabla_1\,f = \vec 0, \\[1.5 ex] \mbox{div\,curl\,grad}\, f = \nabla_3 \circ \nabla_2 \circ \nabla_1\,f = 0, \\[1.5 ex] \mbox{div\,curl\,curl}\, \vec f = \nabla_3 \circ \nabla_2 \circ \nabla_2\,\vec f = 0, \\[1.5 ex] \mbox{grad\,div\,curl}\, \vec f = \nabla_1 \circ \nabla_3 \circ \nabla_2\,\vec f = \vec 0, \\[1.5 ex] \mbox{curl\,grad\,div}\, \vec f = \nabla_2 \circ \nabla_1 \circ \nabla_3\,\vec f = \vec 0. \end{array} \end{equation} If $\mbox{\large \tt f}(k)$ is the number of compositions of the~$k^{\mbox{\scriptsize \rm th}}$~order, then Male\v sevi\' c \cite{HiOrd98} proved \begin{equation} \mbox{\large \tt f}(k) = F_{k+3}, \end{equation} where $F_{k}$ is $k^{\mbox{\scriptsize \rm th}}$ Fibonacci number. \section{The compositions of the differential operations and \textsc{Gateaux} directional derivative of the space~$\mathbb R^{\mbox{\footnotesize \textbf\textit{3}}}$} Let $f \in \mbox{\rm A}_{0}$ be a scalar function and $\vec{e} = (e_1,e_2,e_3)\in \mathbb R^{3}$ be a unit vector. The \textit{Gateaux directional derivative} in direction $\vec{e}$ is defined by \cite[p.$\,$71]{Basov05}$:$ \begin{equation} \mbox{\dire} \, f = \nabla_0 f = \nabla_1 f \cdot \vec{e} = \frac{\partial f}{\partial x_1} \, e_1 + \frac{\partial f}{\partial x_2} \, e_2 + \frac{\partial f}{\partial x_3} \, e_3 : \mbox{\rm A}_{0} \longrightarrow \mbox{\rm A}_{0}. \end{equation} Let us determine the number of meaningful compositions of higher order over the set ${\cal B}_{3} = \{ \nabla_0, \nabla_1, \nabla_2, \nabla_3 \}$. There exist $m=8$ compositions of the second order$:$ \begin{equation} \label{B_Second} \begin{array}{l} \mbox{\dire\,\dire} \, f = \nabla_0 \circ \nabla_0\, f = \nabla_1 {\big (} \, \nabla_1 f \cdot \vec{e} \, {\big )} \cdot \vec{e}, \\[1.25 ex] \mbox{grad\,\dire} \, f = \nabla_1 \circ \nabla_0\, f = \nabla_1 {\big (} \, \nabla_1 f \cdot \vec{e} \, {\big )}, \\[1.25 ex] \Delta f = \mbox{div\,grad} \, f = \nabla_3 \circ \nabla_1 \, f, \\[1.25 ex] \mbox{curl\,curl} \, \vec{f} = \nabla_2 \circ \nabla_2\, \vec{f}, \\[1.25 ex] \mbox{\dire\,div} \, \vec{f} = \nabla_0 \circ \nabla_3\, \vec{f} = {\big (} \nabla_1 \circ \nabla_3 \, \vec{f} \, {\big )} \cdot \vec{e}, \\[1.25 ex] \mbox{grad\,div} \, \vec{f} = \nabla_1 \circ \nabla_3\, \vec{f}, \\[1.25 ex] \mbox{curl\,grad} \, f = \nabla_2 \circ \nabla_1\, f = \vec{0}, \\[1.25 ex] \mbox{div\,curl} \, \vec{f} = \nabla_3 \circ \nabla_2\, \vec{f} = 0; \end{array} \end{equation} and there exist $m=16$ compositions of the third order$:$ \begin{equation} \begin{array}{l} \mbox{\dire\,\dire\,\dire}\, f = \nabla_0 \circ \nabla_0 \circ \nabla_0\, f, \\[1.25 ex] \mbox{grad\,\dire\,\dire}\, f = \nabla_1 \circ \nabla_0 \circ \nabla_0\, f, \\[1.25 ex] \mbox{div\,grad\,\dire}\, f = \nabla_3 \circ \nabla_1 \circ \nabla_0\, f, \\[1.25 ex] \mbox{\dire\,div\,grad}\, f = \nabla_0 \circ \nabla_3 \circ \nabla_1\, f, \\[1.25 ex] \mbox{grad\,div\,grad}\, f = \nabla_1 \circ \nabla_3 \circ \nabla_1\, f, \\[1.25 ex] \mbox{curl\,curl\,curl}\, \vec{f} = \nabla_2 \circ \nabla_2 \circ \nabla_2\, \vec{f}, \\[1.25 ex] \mbox{\dire\,\dire\,div}\, \vec{f} = \nabla_0 \circ \nabla_0 \circ \nabla_3\, \vec{f}, \\[1.25 ex] \mbox{grad\,\dire\,div}\, \vec{f} = \nabla_1 \circ \nabla_0 \circ \nabla_3\, \vec{f}, \\[1.25 ex] \mbox{div\,grad\,div}\, \vec{f} = \nabla_3 \circ \nabla_1 \circ \nabla_3\, \vec{f}, \\[1.25 ex] \mbox{curl\,grad\,\dire}\, f = \nabla_2 \circ \nabla_1 \circ\nabla_0\, \vec{f} = \vec{0}, \\[1.25 ex] \mbox{curl\,curl\,grad}\, f = \nabla_2 \circ \nabla_2 \circ \nabla_1\, f = \vec{0}, \\[1.25 ex] \mbox{div\,curl\,grad}\, f = \nabla_3 \circ \nabla_2 \circ \nabla_1\, f = 0, \\[1.25 ex] \mbox{div\,curl\,curl}\, \vec{f} = \nabla_3 \circ \nabla_2 \circ \nabla_2\, \vec{f} = 0, \\[1.25 ex] \mbox{\dire\,div\,curl}\, \vec{f} = \nabla_0 \circ \nabla_3 \circ \nabla_2\, \vec{f} = 0, \\[1.25 ex] \mbox{grad\,div\,curl}\, \vec{f} = \nabla_1 \circ \nabla_3 \circ \nabla_2\, \vec{f} = \vec{0}, \\[1.25 ex] \mbox{curl\,grad\,div}\, \vec{f} = \nabla_2 \circ \nabla_1 \circ \nabla_3\, \vec{f} = \vec{0}. \end{array} \end{equation} \break \noindent Further on we shall use the method from the paper \cite{HiOrd98}. Let us define a binary relation $\sigma$ ``{\em to be in composition}''$:$ \mbox{$\nabla_{i} \,\sigma\, \nabla_{j}$} iff the composition $\nabla_{j} \circ \nabla_{i}$ is meaningful. Then Cayley table of the relation $\sigma$ is determined by \begin{equation} \label{Cayley_4} \begin{array}{c|cccc} \sigma & \nabla_{0} & \nabla_{1} & \nabla_{2} & \nabla_{3} \\ \hline \nabla_{0} & \top & \top & \bot & \bot \\ \nabla_{1} & \bot & \bot & \top & \top \\ \nabla_{2} & \bot & \bot & \top & \top \\ \nabla_{3} & \top & \top & \bot & \bot \end{array} \end{equation} \medskip \noindent Let us denote by $\nabla_{\!\!-1}$ nowhere-defined function, where domain and range are empty sets \cite{HiOrd96} and let $\nabla_{\!\!-1} \,\sigma\, \nabla_{i}$ hold for $i\!=\!0,1,2,3$. If $G$ is graph which is determined by the relation $\sigma$, then graph of paths of $G$ is the tree with the root $\nabla_{\!\!-1}$ (Fig.~1). %*********************************************************************** Fig 1 ** \hspace*{21.0 mm} %* \setlength{\unitlength}{0.17 cc} %* \begin{picture}(150,50)(0,0) %* \thicklines %* \put(55,40){\circle*{0.8}} %* \put(56,41){\scriptsize$\nabla_{\!\!-1}$} %* \put(130,40){\scriptsize $\mbox{\footnotesize \tt g}(0)=\;1$} %* %* \put(5,30){\line(5,1){50}} %* \put(5,30){\circle*{0.8}} %* \put(3,32){\scriptsize$\nabla_{0}$} %* \put(35,30){\line(2,1){20}} %* \put(35,30){\circle*{0.8}} %* \put(32,32){\scriptsize$\nabla_{1}$} %* \put(75,30){\line(-2,1){20}} %* \put(75,30){\circle*{0.8}} %* \put(75,32){\scriptsize$\nabla_{2}$} %* \put(105,30){\line(-5,1){50}} %* \put(105,30){\circle*{0.8}} %* \put(105,32){\scriptsize$\nabla_{3}$} %* \put(130,30){\scriptsize $\mbox{\footnotesize \tt g}(1)=\;4$} %* \put(-5,20){\line(1,1){10}} %* \put(-5,20){\circle*{0.8}} %* \put(-8,22){\scriptsize$\nabla_{0}$} %* \put(15,20){\line(-1,1){10}} %* \put(15,20){\circle*{0.8}} %* \put(15,22){\scriptsize$\nabla_{1}$} %* \thinlines %* \put(25,20){\line(1,1){10}} %* \put(25,20){\circle*{0.8}} %* \put(22,22){\scriptsize$\nabla_{2}$} %* \thicklines %* \put(45,20){\line(-1,1){10}} %* \put(45,20){\circle*{0.8}} %* \put(43,22){\scriptsize$\nabla_{3}$} %* \put(65,20){\line(1,1){10}} %* \put(65,20){\circle*{0.8}} %* \put(62,22){\scriptsize$\nabla_{2}$} %* \thinlines %* \put(85,20){\line(-1,1){10}} %* \put(85,20){\circle*{0.8}} %* \put(85,22){\scriptsize$\nabla_{3}$} %* \thicklines %* \put(95,20){\line(1,1){10}} %* \put(95,20){\circle*{0.8}} %* \put(92,22){\scriptsize$\nabla_{0}$} %* \put(115,20){\line(-1,1){10}} %* \put(115,20){\circle*{0.8}} %* \put(115,22){\scriptsize$\nabla_{1}$} %* \put(130,20){\scriptsize $\mbox{\footnotesize \tt g}(2)=\;8$} %* %* \put(-7.5,15){\line(1,2){2.5}} %* \put(-2.5,15){\line(-1,2){2.5}} %* \thinlines %* \put(12.5,15){\line(1,2){2.5}} %* \thicklines %* \put(17.5,15){\line(-1,2){2.5}} %* \thinlines %* \put(22.5,15){\line(1,2){2.5}} %* \put(27.5,15){\line(-1,2){2.5}} %* \thicklines %* \put(42.5,15){\line(1,2){2.5}} %* \put(47.5,15){\line(-1,2){2.5}} %* \put(62.5,15){\line(1,2){2.5}} %* \thinlines %* \put(67.5,15){\line(-1,2){2.5}} %* \put(82.5,15){\line(1,2){2.5}} %* \put(87.5,15){\line(-1,2){2.5}} %* \thicklines %* \put(92.5,15){\line(1,2){2.5}} %* \put(97.5,15){\line(-1,2){2.5}} %* \thinlines %* \put(112.5,15){\line(1,2){2.5}} %* \thicklines %* \put(117.5,15){\line(-1,2){2.5}} %* \put(130,15){\scriptsize $\mbox{\footnotesize \tt g}(3)=\;16$} %* \thinlines %* \put(55,4.5){\small Fig. $1$} %* \end{picture} %****************************************************************** \noindent Let $\mbox{\large \tt g}(k)$ be the number of meaningful compositions of the $k^{\mbox{\scriptsize \rm th}}$~order of the functions from ${\cal B}_{3}$ and let $\mbox{\large \tt g}_{i}(k)$ be the number of meaningful compositions of the $k^{\mbox{\scriptsize \rm th}}$~order beginning from the left by $\nabla_{i}$. Then $\mbox{\large \tt g}(k) = \mbox{\large \tt g}_{0}(k) + \mbox{\large \tt g}_{1}(k) + \mbox{\large \tt g}_{2}(k) + \mbox{\large \tt g}_{3}(k)$. Based on the partial self similarity of the tree (Fig.\,$1$) we obtain equalities \begin{equation} \begin{array}{l} \mbox{\large \tt g}_{0}(k) = \mbox{\large \tt g}_{0}(k-1) + \mbox{\large \tt g}_{1}(k-1), \\[1.5 ex] \mbox{\large \tt g}_{1}(k) = \mbox{\large \tt g}_{2}(k-1) + \mbox{\large \tt g}_{3}(k-1), \\[1.5 ex] \mbox{\large \tt g}_{2}(k) = \mbox{\large \tt g}_{2}(k-1) + \mbox{\large \tt g}_{3}(k-1), \\[1.5 ex] \mbox{\large \tt g}_{3}(k) = \mbox{\large \tt g}_{0}(k-1) + \mbox{\large \tt g}_{1}(k-1). \end{array} \end{equation} Hence, the recurrence for $\mbox{\large \tt g}(k)$ is \begin{equation} \mbox{\large \tt g}(k) \!=\! 2 \, \mbox{\large \tt g}(k-1) \end{equation} and because $\mbox{\large \tt g}(1)=4$ we have \begin{equation} \mbox{\large \tt g}(k) = 2^{k+1}. \end{equation} \section{The compositions of differential operations of the space $\mathbb R^{\mbox{\footnotesize \textbf\textit{n}}}$} Let us present the number of meaningful compositions of differential operations in the vector analysis of the space $\mathbb R^{n}$, where differential operations $\nabla_{r}$ $(r \!=\! 1,\ldots,n)$ are defined on corresponding non-empty sets $\mbox{A}_{s}$ $(s \!=\! 1,\ldots,m$ and $m \!=\! \lfloor n/2 \rfloor$, $n \!\geq\! 3)$ according to the papers \cite{HiOrd98}, \cite{HiOrd06}$:$ \vspace*{-5.0 mm} \begin{equation} \label{A_12}{} \begin{tabular}{cc} $\begin{array}{ll} \mbox{\small $\mbox{$\cal A$}_{n}\;(n\!=\!2m)$:}\!\! & \mbox{\small $\nabla_{1}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{2}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{i}$} : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{m}$} : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{n}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{0} \mbox{\normalsize ,} \end{array}$ & $ \begin{array}{ll} \mbox{\small $\mbox{$\cal A$}_{n}\;(n\!=\!2m\!+\!1)$:}\!\! & \mbox{\small $\nabla_{1}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{2}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{i}$} : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{m}$} : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+2}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{n}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{0} \mbox{\normalsize .} \end{array}$ \end{tabular} \end{equation} \vspace*{-1.0 mm} \noindent Let us define {\em higher order differential operations} as meaningful compositions of higher order of differential operations from the set ${\cal A}_{n} = \{\nabla_{1}, \dots, \nabla_{n}\}$. The number of higher order differential operations is given according to the paper \cite{HiOrd98}. Furthermore, let us define a binary relation $\rho$ ``{\em to be in composition}''$:$ $\nabla_{i} \,\rho\, \nabla_{j}$ iff the composition $\nabla_{j} \circ \nabla_{i}$ is meaningful. Then Cayley table of the relation $\rho$ is determined by \begin{equation} \label{Rho} \mbox{\normalsize $\nabla_{i} \,\rho\, \nabla_{j}$} = \left\{ \begin{array}{lll} \top &,& (j = i + 1) \vee (i + j = n + 1); \\[1.0 ex] \bot &,& \mbox{\normalsize otherwise}. \end{array} \right. \end{equation} Let $\mbox{\large \tt A} = [a_{ij}]\in\{\,0,1\}^{n\times n}$ be the adjacency matrix associated with the graph which is determined by the relation $\rho$. Male\v sevi\' c \cite{HiOrd06} proved the following statements. \begin{theo} \label{Th_3_1} Let $P_{n}(\lambda) \!=\! |\mbox{\large \tt A} \!-\! \lambda \mbox{\large \tt I}| \!=\! \alpha_{0} \lambda^{n} + \alpha_{1} \lambda^{n-1} + \dots + \alpha_{n}$ be the characteristic polynomial of the matrix $\mbox{\large \tt A}$ and $v_{n} = [ \, 1 \, \dots \, 1 \, ]_{1 \times n}$. If $\mbox{\large \tt f}(k)$ is the number of the $k^{\it \footnotesize th}\!$~order differential operations, then the following formulas hold: \begin{equation} \mbox{\large \tt f}(k) = v_n \cdot \mbox{\large \tt A}^{k-1} \cdot v^{T}_n \end{equation} and \begin{equation} \alpha_{0} \mbox{\large \tt f}(k) + \alpha_{1} \mbox{\large \tt f}(k-1) + \dots + \alpha_{n} \mbox{\large \tt f}(k-n) = 0 \quad (k > n). \end{equation} \end{theo} \break \begin{lemma} \label{lemma_stara_1} Let $P_{n}(\lambda)$ be the characteristic polynomial of the matrix $\mbox{\large \tt A}$. Then the following recurrence holds: \begin{equation} \label{lemma_stara_1_Form_1} P_{n}(\lambda) = \lambda^2 {\big (} P_{n-2}(\lambda) - P_{n-4}(\lambda) {\big )}. \end{equation} \end{lemma} \begin{lemma} \label{lemma_stara_2} Let $P_{n}(\lambda)$ be the characteristic polynomial of the matrix $\mbox{\large \tt A}$. Then it has the following explicit form: \begin{equation} \label{lemma_stara_2_Form_2} \quad P_{n}(\lambda) = \left\{ \begin{array}{ccl} \displaystyle\sum\limits_{k=1}^{\lfloor \frac{n+2}{4} \rfloor +1}{(-1)^{k-1} { \: \mbox{\scriptsize $\displaystyle\frac{n}{2}\!-\!k\!+\!2$} \: \choose \: \mbox{\scriptsize $k\!-\!1$} \: } \lambda^{n-2k+2}} \!\!&\!\!,\!\!& n\!=\!2m; \\[2.0 ex] \!\!\!\displaystyle\sum\limits_{k=1}^{\lfloor \frac{n+2}{4} \rfloor +2}{\!\!\!\!(-1)^{k-1}\!{\Bigg (} \!{ \: \mbox{\scriptsize $\displaystyle\frac{n\!+\!3}{2}\!-\!k$} \: \choose \: \mbox{\scriptsize $k\!-\!1$} \: } \!+\! { \: \mbox{\scriptsize $\displaystyle\frac{n\!+\!3}{2}\!-\!k$} \: \choose \: \mbox{\scriptsize $k\!-\!2$} \: } \! \lambda \! {\Bigg )} \lambda^{n-2k+2}} \!&\!\!,\!\!& n\!=\!2m\!+\!1.\!\!\!\! \end{array} \right. \end{equation} \end{lemma} \noindent From previous statements one can obtain the recurrences in the table, \cite{HiOrd98}: { \footnotesize \begin{center} \begin{tabular}{|c|c|} \hline {\small \rm Dimension} & {\small \rm \quad Recurrence for the number of the $k^{\mbox{\scriptsize \rm th}}$ order differential operations \quad\quad} \\ \hline $n = \;$ 3 & $ \mbox{\normalsize \tt f}(k) = \mbox{\normalsize \tt f}(k-1) + \mbox{\normalsize \tt f}(k-2)$ \\ \hline $n = \;$ 4 & $\mbox{\normalsize \tt f}(k) = 2 \mbox{\normalsize \tt f}(k-2)$ \\ \hline $n = \;$ 5 & $\mbox{\normalsize \tt f}(k) = \mbox{\normalsize \tt f}(k-1) + 2 \mbox{\normalsize \tt f}(k-2) - \mbox{\normalsize \tt f}(k-3)$ \\ \hline $n = \;$ 6 & $\mbox{\normalsize \tt f}(k) = 3 \mbox{\normalsize \tt f}(k-2) - \mbox{\normalsize \tt f}(k-4)$ \\ \hline $n = \;$ 7 & $\mbox{\normalsize \tt f}(k) = \mbox{\normalsize \tt f}(k-1) + 3 \mbox{\normalsize \tt f}(k-2) - 2 \mbox{\normalsize \tt f}(k-3) - \mbox{\normalsize \tt f}(k-4)$ \\ \hline $n = \;$ 8 & $\mbox{\normalsize \tt f}(k) = 4 \mbox{\normalsize \tt f}(k-2) - 3 \mbox{\normalsize \tt f}(k-4)$ \\ \hline $n = \;$ 9 & $\mbox{\normalsize \tt f}(k) = \mbox{\normalsize \tt f}(k-1) + 4 \mbox{\normalsize \tt f}(k-2) - 3 \mbox{\normalsize \tt f}(k-3) - 3 \mbox{\normalsize \tt f}(k-4) + \mbox{\normalsize \tt f}(k-5)$ \\ \hline $n = $ 10& $\mbox{\normalsize \tt f}(k) = 5 \mbox{\normalsize \tt f}(k-2) - 6 \mbox{\normalsize \tt f}(k-4) + \mbox{\normalsize \tt f}(k-6)$ \\ \hline \end{tabular} \end{center} } \smallskip \noindent The values of the function $\mbox{\large \tt f}(k)$, for small values of the argument $k$, are given in the database of integer sequences \cite{Sloane07} as the following sequences \seqnum{A020701} $(n=3)$, \seqnum{A090989} $(n=4)$, \seqnum{A090990} $(n=5)$, \seqnum{A090991} $(n=6)$, \seqnum{A090992} $(n=7)$, \seqnum{A090993} $(n=8)$, \seqnum{A090994} $(n=9)$, \seqnum{A090995} $(n=10)$. \section{The compositions of differential operations and Gateaux directional derivative of the space $\mathbb R^{\mbox{\footnotesize \textbf\textit{n}}}$} Let $f \in \mbox{\rm A}_{0}$ be a scalar function and $\vec{e} = (e_1,\dots,e_n) \in \mathbb R^{n}$ be a unit vector. The \textit{Gateaux directional derivative} in direction $\vec{e}$ is defined by \cite[p.$\,$71]{Basov05}$:$ \begin{equation} \mbox{\dire} \, f = \nabla_0 f = \displaystyle \sum\limits_{k=1}^{n}{ \frac{\partial f}{\partial x_k} \, e_k} : \mbox{\rm A}_{0} \longrightarrow \mbox{\rm A}_{0}. \end{equation} Let us extend the set of differential operations ${\cal A}_{n} = \{ \nabla_{1},\dots,\nabla_{n} \}$ with Gateaux directional derivative to the set ${\cal B}_{n} = {\cal A}_{n} \cup \{ \nabla_{0} \} = \{ \nabla_{0},\nabla_{1},\dots,\nabla_{n} \}$$:$ \begin{equation} \label{B_12} \mbox{$\begin{array}{ll} \mbox{\small $\mbox{$\cal B$}_{n}\;(n\!=\!2m)$:}\!\! & \mbox{\small $\nabla_{0}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{0} \\ & \mbox{\small $\nabla_{1}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{2}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{i}$} : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{m}$} : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{n}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{0} \mbox{\normalsize ,} \end{array} $} \quad \mbox{$ \begin{array}{ll} \mbox{\small $\mbox{$\cal B$}_{n}\;(n\!=\!2m\!+\!1)$:}\!\! & \mbox{\small $\nabla_{0}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{0} \\ & \mbox{\small $\nabla_{1}$} : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{2}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{i}$} : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{m}$} : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m} \\ & \mbox{\small $\nabla_{m+2}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\ & \,\,\vdots \\ & \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1} \\ & \mbox{\small $\nabla_{n}$} : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{0} \mbox{\normalsize .} \end{array} $} \end{equation} Let us define \textit{higher order differential operations with Gateaux derivative} as the meaningful compositions of higher order of the functions from the set ${\cal B}_{n} = \{ \nabla_{0},\nabla_{1},\dots, \nabla_{n} \}$. Our aim is to determine the number of higher order differential operations with Gateaux derivative. Let us define a binary relation~$\sigma$ ``{\em to~be~in~composition}''$:$ \begin{equation} \label{Rho'2} \nabla_{i} \,\sigma\, \nabla_{j} = \left\{ \begin{array}{lll} \top\! &\!\!,\!\!& (i\!=\!0 \wedge j\!=\!0) \vee (i\!=\!n \wedge j\!=\!0) \vee (j\!=\!i\!+\!1) \vee (i\!+\!j\!=\!n\!+\!1); \\[1.0 ex] \bot\! &\!\!,\!\!& \mbox{\normalsize otherwise}. \end{array} \right. \end{equation} and let $\mbox{\large \tt B} = [b_{ij}]\in\{\,0,1\}^{(n+1) \times n}$ be the adjacency matrix associated with the graph which is determined by relation $\sigma$. So, analogously to the paper \cite{HiOrd06}, the following statements hold. \begin{theo} \label{Th_4_1} Let $Q_{n}(\lambda) \!=\! |\mbox{\large \tt B} \!- \! \lambda \mbox{\large \tt I}| = \beta_{0} \lambda^{n+1} + \beta_{1}\lambda^{n} + \dots + \beta_{n+1}$ be the characteristic polynomial of the matrix $\mbox{\large \tt B}$ and $v_{n+1} = [ \, 1 \, \dots \, 1 \, ]_{1 \times (n+1)}$. If $\mbox{\large \tt g}(k)$ is the number of the $k^{\it \footnotesize th}\!$~order differential operations with Gateaux derivative, then the following formulas hold: \begin{equation} \mbox{\large \tt g}(k) = v_{n+1} \cdot \mbox{\large \tt B}^{k-1} \cdot v^{T}_{n+1} \end{equation} and \begin{equation} \beta_{0} \mbox{\large \tt g}(k) + \beta_{1} \mbox{\large \tt g}(k-1) + \dots + \beta_{n+1} \mbox{\large \tt g}(k-(n+1)) = 0 \quad (k > n\!+\!1). \end{equation} \end{theo} \begin{lemma} \label{lemma_4_2} Let $Q_{n}(\lambda)$ and $P_{n}(\lambda)$ be the characteristic polynomials of the matrices $\mbox{\large \tt B}$ and $\mbox{\large \tt A}$ respectively. Then the following equality holds: \begin{equation} \label{veza izmedju A i B} Q_{n}(\lambda) = \lambda^2 P_{n-2}(\lambda) - \lambda P_{n}(\lambda). \end{equation} \end{lemma} \begin{proof} Let us calculate the characteristic polynomial \begin{equation} Q_{n}(\lambda) = |\mbox{\large \tt B} - \lambda \mbox{\large \tt I}| = \mbox{\footnotesize $\left| \begin{array}{rrrrrrrrr} 1-\lambda & 1 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & -\lambda & 1 & 0 & \dots & 0 & 0 & 0 & 1 \\ 0 & 0 & -\lambda & 1 & \dots & 0 & 0 & 1 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 1 & \dots & 0 & -\lambda & 1 & 0 \\ 0 & 0 & 1 & 0 & \dots & 0 & 0 & -\lambda & 1 \\ 1 & 1 & 0 & 0 & \dots & 0 & 0 & 0 & -\lambda \end{array} \right|$}\,. \end{equation} \medskip \noindent Expanding the determinant $Q_{n}(\lambda)$ by the first column we have \begin{equation} \label{jed. 1} Q_{n}(\lambda) = (1-\lambda) P_{n}(\lambda) + (-1)^{n+2} D_{n}(\lambda), \end{equation} where \begin{equation} \label{detD 1} D_{n}(\lambda) = \mbox{\footnotesize $\left| \begin{array}{rrrrrrrrr} 1 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ -\lambda & 1 & 0 & 0 & \dots & 0 & 0 & 0 & 1 \\ 0 & -\lambda & 1 & 0 & \dots & 0 & 0 & 1 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 1 & \dots & -\lambda & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & \dots & 0 & -\lambda & 1 & 0 \\ 0 & 1 & 0 & 0 & \dots & 0 & 0 & -\lambda & 1 \end{array} \right|$}\,. \end{equation} \medskip \noindent Let us expand the determinant $D_{n}(\lambda)$ by the first row and then in the next step, multiply the first row by $-1$ and add it to the last row. We obtain the determinant of order $n-1:$ \begin{equation} \label{detD 2} D_{n}(\lambda) = \mbox{\footnotesize $\left| \begin{array}{rrrrrrrrr} 1 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & 1 \\ -\lambda & 1 & 0 & 0 & \dots & 0 & 0 & 1 & 0 \\ 0 & -\lambda & 1 & 0 & \dots & 0 & 1 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 1 & 0 & \dots & -\lambda & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & \dots & 0 & -\lambda & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & -\lambda & 0 \end{array} \right|$}\,. \end{equation} \medskip \noindent Expanding the previous determinant by the last column we have \begin{equation} \label{detD 3} D_{n}(\lambda) = (-1)^{n} \mbox{\footnotesize $\left| \begin{array}{rrrrrrrrr} -\lambda & 1 & 0 & 0 & \dots & 0 & 0 & 0 & 1 \\ 0 & -\lambda & 1 & 0 & \dots & 0 & 0 & 1 & 0 \\ 0 & 0 & -\lambda & 1 & \dots & 0 & 1 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 1 & 0 & \dots & 0 & -\lambda & 1 & 0 \\ 0 & 1 & 0 & 0 & \dots & 0 & 0 & -\lambda & 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & -\lambda \end{array} \right|$}\,. \end{equation} \break \noindent If we expand the previous determinant by the last row and if we expand the obtained determinant by the first column, we have the determinant of order $n-4:$ \begin{equation} \label{detD 4} D_{n}(\lambda) = (-1)^{n}\lambda^2 \mbox{\footnotesize $\left| \begin{array}{rrrrrrrrr} -\lambda & 1 & 0 & 0 & \dots & 0 & 0 & 0 & 1 \\ 0 & -\lambda & 1 & 0 & \dots & 0 & 0 & 1 & 0 \\ 0 & 0 & -\lambda & 1 & \dots & 0 & 1 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots &\vdots & \vdots \\ 0 & 0 & 1 & 0 & \dots & 0 & -\lambda & 1 & 0 \\ 0 & 1 & 0 & 0 & \dots & 0 & 0 & -\lambda & 1 \\ 1 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & -\lambda \end{array} \right|$}\,. \end{equation} \medskip \noindent In other words \begin{equation} \label{jedD 1} D_{n}(\lambda) =(-1)^{n}\lambda^2P_{n-4}(\lambda) . \end{equation} \bigskip \noindent From equalities {\rm (\ref{jedD 1})} and {\rm (\ref{jed. 1})} there follows$:$ \begin{equation} Q_{n}(\lambda) =(1-\lambda) P_{n}(\lambda) + \lambda^2P_{n-4}(\lambda) . \end{equation} \noindent On the basis of Lemma {\rm \ref{lemma_stara_1}.} the following equality holds: \begin{equation} Q_{n}(\lambda) = \lambda^2 P_{n-2}(\lambda) -\lambda P_{n}(\lambda) . \end{equation} \end{proof} \begin{lemma} \label{lemma_4_3} Let $Q_{n}(\lambda)$ be the characteristic polynomial of the matrix $\mbox{\large \tt B}$. Then the following recurrence holds: \begin{equation} Q_{n}(\lambda) = \lambda^2 {\big (} Q_{n-2}(\lambda) - Q_{n-4}(\lambda) {\big )}. \end{equation} \end{lemma} \begin{proof} On the basis of Lemma {\rm \ref{lemma_stara_1}.} and Lemma {\rm \ref{lemma_4_2}.} the Lemma follows. \end{proof} \bigskip \begin{lemma} \label{lemma_4_4} Let $Q_{n}(\lambda)$ be the characteristic polynomial of the matrix $\mbox{\large \tt B}$. Then it has the following explicit form$:$ \begin{equation} \label{P_explicit} \quad Q_{n}(\lambda) = \left\{ \begin{array}{ccl} (\lambda-2)\displaystyle\sum\limits_{k=1}^{\lfloor \frac{n\!\!}{\,4} \rfloor +1}{(-1)^{k-1} {\:\mbox{\scriptsize $\displaystyle\frac{n+1}{2}\!-\!k$}\: \choose \:\mbox{\scriptsize $k\!-\!1$}\:} \lambda^{n-2k+2}} \!\!&\!\!,\!\!& n\!=\!2m\!+\!1; \\[3.0 ex] \!\!\!\displaystyle\sum\limits_{k=1}^{\lfloor \frac{n+3}{4} \rfloor +2}{\!\!\!\!(-1)^{k-1}\!{\Bigg (} \!{\:\mbox{\scriptsize $\displaystyle\frac{n}{2}\!-\!k\!+\!2$}\: \choose \mbox{\scriptsize $k\!-\!1$}} \!+\! {\:\mbox{\scriptsize $\displaystyle\frac{n}{2}\!-\!k\!+\!2$}\: \choose \!\!\mbox{\scriptsize $k\!-\!2$}\:} \! \lambda \! {\Bigg )} \lambda^{n-2k+3}} \!&\!\!,\!\!& n\!=\!2m.\!\!\!\! \end{array} \right. \end{equation} \end{lemma} \begin{proof} On the basis of Lemma {\rm \ref{lemma_stara_2}} and Lemma {\rm \ref{lemma_4_2}.} the Lemma follows. \end{proof} The recurrences for dimensions $n \!=\! 3,4, \dots, 10\,$ are obtained by means of Male\v sevi\'c-Jovovi\'c \cite{PowerMatrix} and they are given in the table below. {\footnotesize \begin{center} \begin{tabular}{|c|c|} \hline $\!\!${\small \rm Dimension}$\!\!$ & $\!\!${\small \rm Recurrence$\!$ for$\!$ the$\!$ num.$\!$ of$\!$ the $\!k^{\mbox{\scriptsize \rm th}}\!$ order$\!$ diff.$\!$ operations$\!$ with$\!$ Gateaux$\!$ derivative}$\!\!$ \\ \hline $n = \;$ 3 & $\mbox{\normalsize \tt g}(k) = 2 \mbox{\normalsize \tt g}(k-1)$ \\ \hline $n = \;$ 4 & $\mbox{\normalsize \tt g}(k) = \mbox{\normalsize \tt g}(k-1) + 2 \mbox{\normalsize \tt g}(k-2) - \mbox{\normalsize \tt g}(k-3)$ \\ \hline $n = \;$ 5 & $\mbox{\normalsize \tt g}(k) = 2 \mbox{\normalsize \tt g}(k-1) + \mbox{\normalsize \tt g}(k-2) - 2 \mbox{\normalsize \tt g}(k-3)$ \\ \hline $n = \;$ 6 & $\mbox{\normalsize \tt g}(k) = \mbox{\normalsize \tt g}(k-1) + 3 \mbox{\normalsize \tt g}(k-2) - 2 \mbox{\normalsize \tt g}(k-3) - \mbox{\normalsize \tt g}(k-4)$ \\ \hline $n = \;$ 7 & $\mbox{\normalsize \tt g}(k) = 2 \mbox{\normalsize \tt g}(k-1) + 2 \mbox{\normalsize \tt g}(k-2) - 4 \mbox{\normalsize \tt g}(k-3)$ \\ \hline $n = \;$ 8 & $\mbox{\normalsize \tt g}(k) = \mbox{\normalsize \tt g}(k-1) + 4 \mbox{\normalsize \tt g}(k-2) - 3 \mbox{\normalsize \tt g}(k-3) - 3 \mbox{\normalsize \tt g}(k-4) + \mbox{\normalsize \tt g}(k-5)$ \\ \hline $n = \;$ 9 & $\mbox{\normalsize \tt g}(k) = 2 \mbox{\normalsize \tt g}(k-1) + 3 \mbox{\normalsize \tt g}(k-2) - 6 \mbox{\normalsize \tt g}(k-3) - \mbox{\normalsize \tt g}(k-4) + 2 \mbox{\normalsize \tt g}(k-5)$ \\ \hline $n = $10 & $\mbox{\normalsize \tt g}(k) = \mbox{\normalsize \tt g}(k-1) + 5 \mbox{\normalsize \tt g}(k-2) - 4 \mbox{\normalsize \tt g}(k-3) - 6 \mbox{\normalsize \tt g}(k-4) + 3 \mbox{\normalsize \tt g}(k-5) + \mbox{\normalsize \tt g}(k-6)$ \\ \hline \end{tabular} \end{center}} \smallskip \noindent The values of the function $\mbox{\large \tt g}(k)$, for small values of the argument $k$, are given in the database of integer sequences \cite{Sloane07} as the following sequences \seqnum{A000079} $(n=3)$, \seqnum{A090990} $(n=4)$, \seqnum{A007283} $(n=5)$, \seqnum{A090992} $(n=6)$, \seqnum{A000079} $(n=7)$, \seqnum{A090994} $(n=8)$, \seqnum{A020714} $(n=9)$, \seqnum{A129638} $(n=10)$. \bibstyle{plain} \begin{thebibliography}{99} \bibitem{Ivanov02} Ivanov, A. B., Vector analysis, in M. Hazewinkel, ed., {\it Encyclopaedia of Mathematics}, Springer, 2002. Text available at \href{http://eom.springer.de/V/v096360.htm}{\tt http://eom.springer.de/V/v096360.htm}. \bibitem{Korn00} G. A. Korn and T. M. Korn, {\it Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review}, Courier Dover Publications, 2000. \bibitem{HiOrd96} B. J. Male\v sevi\' c, A note on higher-order differential operations, {\it Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat.} {\bf 7} (1996), 105--109. Text available at \href{http://pefmath2.etf.bg.ac.yu/files/116/846.pdf}{\tt http://pefmath2.etf.bg.ac.yu/files/116/846.pdf}. \bibitem{HiOrd98} B. J. Male\v sevi\' c, Some combinatorial aspects of differential operation composition on the space $\mbox{ R}^{n}$, {\it Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat.} {\bf 9} (1998), 29--33. Text available at \href{http://pefmath2.etf.bg.ac.yu/files/118/869.pdf}{\tt http://pefmath2.etf.bg.ac.yu/files/118/869.pdf}. \bibitem{Basov05} S. Basov, {\it Multidimensional Screening}, Springer 2005. \bibitem{HiOrd06} B. J. Male\v sevi\'c, Some combinatorial aspects of the composition of a set of function, {\it Novi Sad J. Math.}, {\bf 36} (1), 2006, 3--9. Text available at \href{http://www.im.ns.ac.yu/NSJOM/Papers/36_1/NSJOM_36_1_003_009.pdf}{ \tt http://www.im.ns.ac.yu/NSJOM/Papers/36\underline{\,\,}1/NSJOM\underline{\,\,}36\underline{\,\,}1\underline{\,\,}003\underline{\,\,}009.pdf}. \bibitem{PowerMatrix} B. J. Male\v sevi\'c and I. V. Jovovi\'c, {\it A procedure for finding the $k^{th}$ power of a matrix}. \href{http://www.maplesoft.com/Applications}{\tt http://www.maplesoft.com/Applications}. \bibitem{Sloane07} N. J. A. Sloane, {\it The On-Line Encyclopedia of Integer Sequences}. \href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/\~{}njas/sequences/}. \end{thebibliography} \vfill\eject \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: 05C30, 26B12, 58C20. \\ \noindent \emph{Keywords:} compositions of the differential operations, enumeration of graphs and maps, Gateaux directional derivative. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000079}, \seqnum{A007283}, \seqnum{A020701}, \seqnum{A020714}, \seqnum{A090989}, \seqnum{A090990}, \seqnum{A090991}, \seqnum{A090992}, \seqnum{A090993}, \seqnum{A090994}, \seqnum{A090995}, and \seqnum{A129638}.) \bigskip\hrule\bigskip \vspace*{+.1in} \noindent Received June 5 2007; revised version received July 30 2007. Published in {\it Journal of Integer Sequences}, August 3 2007. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}. \end{document}