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\newtheorem{lemma}[theo]{Lemma}                                                                             
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\begin{document}

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\epsfxsize=4in \leavevmode\epsffile{logo129.eps}             
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\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 &amp; \nabla_{0} &amp; \nabla_{1} &amp; \nabla_{2} &amp; \nabla_{3}  \\ \hline
\nabla_{0} &amp; \top &amp; \top       &amp; \bot       &amp; \bot        \\
\nabla_{1} &amp; \bot &amp; \bot       &amp; \top       &amp; \top        \\
\nabla_{2} &amp; \bot &amp; \bot       &amp; \top       &amp; \top        \\
\nabla_{3} &amp; \top &amp; \top       &amp; \bot       &amp; \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 **
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\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}}                                                  %*
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\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$\nabla_{1}$}                                           %*
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\put(130,20){\scriptsize $\mbox{\footnotesize \tt g}(2)=\;8$}                  %*
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\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$}_{n}\;(n\!=\!2m)$:}\!\!
           &amp; \mbox{\small $\nabla_{1}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1}   \\
           &amp; \mbox{\small $\nabla_{2}$}   : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2}   \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{i}$}   : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{m}$}   : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\
           &amp; \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1}   \\
           &amp; \mbox{\small $\nabla_{n}$}   : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{0}
             \mbox{\normalsize ,}
\end{array}$
           &amp;
$ \begin{array}{ll}
\mbox{\small $\mbox{$\cal A$}_{n}\;(n\!=\!2m\!+\!1)$:}\!\!
           &amp; \mbox{\small $\nabla_{1}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1}   \\
           &amp; \mbox{\small $\nabla_{2}$}   : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2}   \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{i}$}   : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{m}$}   : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\
           &amp; \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m}   \\
           &amp; \mbox{\small $\nabla_{m+2}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\
           &amp; \,\,\vdots                                                                 \\
           &amp; \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1}   \\
           &amp; \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 &amp;,&amp; (j = i + 1)    \vee   (i + j = n + 1);                                 \\[1.0 ex]
\bot &amp;,&amp; \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 &gt; 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}}
\!\!&amp;\!\!,\!\!&amp; 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}}
\!&amp;\!\!,\!\!&amp; 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}
 &amp;     {\small \rm \quad Recurrence for the number of the $k^{\mbox{\scriptsize \rm th}}$ order differential operations \quad\quad} \\ \hline
$n = \;$ 3 &amp; $ \mbox{\normalsize \tt f}(k)
             = \mbox{\normalsize \tt f}(k-1)
             + \mbox{\normalsize \tt f}(k-2)$                                         \\ \hline
$n = \;$ 4 &amp; $\mbox{\normalsize \tt f}(k)
             = 2 \mbox{\normalsize \tt f}(k-2)$                                       \\ \hline
$n = \;$ 5 &amp; $\mbox{\normalsize \tt f}(k)
             = \mbox{\normalsize \tt f}(k-1)
             + 2 \mbox{\normalsize \tt f}(k-2)
             - \mbox{\normalsize \tt f}(k-3)$                                         \\ \hline
$n = \;$ 6 &amp; $\mbox{\normalsize \tt f}(k)
             = 3 \mbox{\normalsize \tt f}(k-2)
             - \mbox{\normalsize \tt f}(k-4)$                                         \\ \hline
$n = \;$ 7 &amp; $\mbox{\normalsize \tt f}(k)
             = \mbox{\normalsize \tt f}(k-1)
             + 3 \mbox{\normalsize \tt f}(k-2)
             - 2 \mbox{\normalsize \tt f}(k-3)
             - \mbox{\normalsize \tt f}(k-4)$                                       \\ \hline
$n = \;$ 8 &amp; $\mbox{\normalsize \tt f}(k)
             = 4 \mbox{\normalsize \tt f}(k-2)
             - 3 \mbox{\normalsize \tt f}(k-4)$                                     \\ \hline
$n = \;$ 9 &amp; $\mbox{\normalsize \tt f}(k)
             = \mbox{\normalsize \tt f}(k-1)
             + 4 \mbox{\normalsize \tt f}(k-2)
             - 3 \mbox{\normalsize \tt f}(k-3)
             - 3 \mbox{\normalsize \tt f}(k-4)
             + \mbox{\normalsize \tt f}(k-5)$                                      \\ \hline
$n =   $ 10&amp; $\mbox{\normalsize \tt f}(k)
             = 5 \mbox{\normalsize \tt f}(k-2)
             - 6 \mbox{\normalsize \tt f}(k-4)
             + \mbox{\normalsize \tt f}(k-6)$                                      \\ \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$}_{n}\;(n\!=\!2m)$:}\!\!
             &amp; \mbox{\small $\nabla_{0}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{0}   \\
             &amp; \mbox{\small $\nabla_{1}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1}   \\
             &amp; \mbox{\small $\nabla_{2}$}   : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2}   \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{i}$}   : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{m}$}   : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\
             &amp; \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1}   \\
             &amp; \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$}_{n}\;(n\!=\!2m\!+\!1)$:}\!\!
             &amp; \mbox{\small $\nabla_{0}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{0}   \\
             &amp; \mbox{\small $\nabla_{1}$}   : \mbox{A}_{0} \!\rightarrow\! \mbox{A}_{1}   \\
             &amp; \mbox{\small $\nabla_{2}$}   : \mbox{A}_{1} \!\rightarrow\! \mbox{A}_{2}   \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{i}$}   : \mbox{A}_{i-1} \!\rightarrow\! \mbox{A}_{i} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{m}$}   : \mbox{A}_{m-1} \!\rightarrow\! \mbox{A}_{m} \\
             &amp; \mbox{\small $\nabla_{m+1}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m}   \\
             &amp; \mbox{\small $\nabla_{m+2}$} : \mbox{A}_{m} \!\rightarrow\! \mbox{A}_{m-1} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{n-j}$} : \mbox{A}_{j+1} \!\rightarrow\! \mbox{A}_{j} \\
             &amp; \,\,\vdots                                                                 \\
             &amp; \mbox{\small $\nabla_{n-1}$} : \mbox{A}_{2} \!\rightarrow\! \mbox{A}_{1}   \\
             &amp; \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\! &amp;\!\!,\!\!&amp; (i\!=\!0 \wedge j\!=\!0) \vee (i\!=\!n \wedge j\!=\!0)
\vee (j\!=\!i\!+\!1) \vee (i\!+\!j\!=\!n\!+\!1);               \\[1.0 ex]
\bot\! &amp;\!\!,\!\!&amp; \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 &gt; 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 $\left|
\begin{array}{rrrrrrrrr}
1-\lambda &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; -\lambda  &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\
0 &amp; 0 &amp; -\lambda  &amp; 1 &amp; \dots &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;
\vdots &amp; \vdots &amp; \vdots &amp; \vdots             \\
0 &amp; 0 &amp; 0 &amp; 1 &amp; \dots &amp; 0 &amp; -\lambda &amp; 1 &amp; 0  \\
0 &amp; 0 &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; -\lambda &amp; 1  \\
1 &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; -\lambda
\end{array}
\right|$}\,.
\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 $\left|
\begin{array}{rrrrrrrrr}
1        &amp; 0        &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
-\lambda &amp; 1        &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\
0        &amp; -\lambda &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;
\vdots &amp; \vdots &amp; \vdots &amp; \vdots            \\
0 &amp; 0 &amp; 0 &amp; 1 &amp; \dots &amp; -\lambda &amp; 1 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; -\lambda &amp; 1 &amp; 0 \\
0 &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; -\lambda &amp; 1
\end{array}
\right|$}\,.
\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 $\left|
\begin{array}{rrrrrrrrr}
       1 &amp; 0 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\
-\lambda &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; -\lambda &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;
\vdots &amp; \vdots &amp; \vdots &amp; \vdots            \\
0 &amp; 0 &amp; 1 &amp; 0 &amp; \dots &amp; -\lambda &amp; 1 &amp; 0 &amp; 0 \\
0 &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; -\lambda &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; -\lambda &amp; 0
\end{array}
\right|$}\,.
\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 $\left|
\begin{array}{rrrrrrrrr}
-\lambda &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 1  \\
0 &amp; -\lambda &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 1 &amp; 0  \\
0 &amp; 0 &amp; -\lambda &amp; 1 &amp; \dots &amp; 0 &amp; 1 &amp; 0 &amp; 0  \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;
\vdots &amp; \vdots &amp; \vdots &amp; \vdots             \\
0 &amp; 0 &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; -\lambda &amp; 1 &amp; 0  \\
0 &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; -\lambda &amp; 1  \\
0 &amp; 0 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; -\lambda
\end{array}
\right|$}\,.
\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 $\left|
\begin{array}{rrrrrrrrr}
-\lambda &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; 1  \\
0 &amp; -\lambda &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 1 &amp; 0  \\
0 &amp; 0 &amp; -\lambda &amp; 1 &amp; \dots &amp; 0 &amp; 1 &amp; 0 &amp; 0  \\
\vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;
\vdots &amp; \vdots &amp;\vdots &amp; \vdots              \\
0 &amp; 0 &amp; 1 &amp; 0 &amp; \dots &amp; 0 &amp; -\lambda &amp; 1 &amp; 0  \\
0 &amp; 1 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; -\lambda &amp; 1  \\
1 &amp; 0 &amp; 0 &amp; 0 &amp; \dots &amp; 0 &amp; 0 &amp; 0 &amp; -\lambda
\end{array}
\right|$}\,.
\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}}
\!\!&amp;\!\!,\!\!&amp; 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}}
\!&amp;\!\!,\!\!&amp; 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}$\!\!$
&amp;     $\!\!${\small \rm Recurrence$\!$ for$\!$ the$\!$ num.$\!$ of$\!$ the $\!k^{\mbox{\scriptsize \rm th}}\!$ order$\!$ diff.$\!$ operations$\!$
                        with$\!$ Gateaux$\!$ derivative}$\!\!$     \\ \hline
$n = \;$ 3 &amp; $\mbox{\normalsize \tt g}(k)
             = 2 \mbox{\normalsize \tt g}(k-1)$                                       \\ \hline
$n = \;$ 4 &amp; $\mbox{\normalsize \tt g}(k)
             = \mbox{\normalsize \tt g}(k-1)
             + 2 \mbox{\normalsize \tt g}(k-2)
             - \mbox{\normalsize \tt g}(k-3)$                                         \\ \hline
$n = \;$ 5 &amp; $\mbox{\normalsize \tt g}(k)
             = 2 \mbox{\normalsize \tt g}(k-1)
             + \mbox{\normalsize \tt g}(k-2)
             - 2 \mbox{\normalsize \tt g}(k-3)$                                       \\ \hline
$n = \;$ 6 &amp; $\mbox{\normalsize \tt g}(k)
             = \mbox{\normalsize \tt g}(k-1)
             + 3 \mbox{\normalsize \tt g}(k-2)
             - 2 \mbox{\normalsize \tt g}(k-3)
             - \mbox{\normalsize \tt g}(k-4)$                                         \\ \hline
$n = \;$ 7 &amp; $\mbox{\normalsize \tt g}(k)
             = 2 \mbox{\normalsize \tt g}(k-1)
             + 2 \mbox{\normalsize \tt g}(k-2)
             - 4 \mbox{\normalsize \tt g}(k-3)$                                       \\ \hline
$n = \;$ 8 &amp; $\mbox{\normalsize \tt g}(k)
             = \mbox{\normalsize \tt g}(k-1)
             + 4 \mbox{\normalsize \tt g}(k-2)
             - 3 \mbox{\normalsize \tt g}(k-3)
             - 3 \mbox{\normalsize \tt g}(k-4)
             + \mbox{\normalsize \tt g}(k-5)$                                         \\ \hline
$n = \;$ 9 &amp; $\mbox{\normalsize \tt g}(k)
             = 2 \mbox{\normalsize \tt g}(k-1)
             + 3 \mbox{\normalsize \tt g}(k-2)
             - 6 \mbox{\normalsize \tt g}(k-3)
             - \mbox{\normalsize \tt g}(k-4)
             + 2 \mbox{\normalsize \tt g}(k-5)$                                       \\ \hline
$n =   $10 &amp; $\mbox{\normalsize \tt g}(k)
             = \mbox{\normalsize \tt g}(k-1)
             + 5 \mbox{\normalsize \tt g}(k-2)
             - 4 \mbox{\normalsize \tt g}(k-3)
             - 6 \mbox{\normalsize \tt g}(k-4)
             + 3 \mbox{\normalsize \tt g}(k-5)
             + \mbox{\normalsize \tt g}(k-6)$                                        \\ \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}

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{\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}

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\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.

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\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.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.

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