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\vskip 1cm{\LARGE\bf 
Translated Whitney and $r$-Whitney Numbers: \\
\vskip .12 in
A Combinatorial Approach 
}
\vskip 1cm
\large
Hac\`{e}ne Belbachir and Imad Eddine Bousbaa \\
USTHB \\
Faculty of Mathematics\\
RECITS Laboratory, DG-RSDT\\
BP 32, El Alia 16111 \\
Bab Ezzouar, Algiers\\
Algeria\\
\href{mailto:hbelbachir@usthb.dz}{\tt hbelbachir@usthb.dz} \\
\href{mailto:hacenebelbachir@gmail.com}{\tt hacenebelbachir@gmail.com} \\
\href{mailto:bousbaa.imadeddine@gmail.com}{\tt bousbaa.imadeddine@gmail.com}\\
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\begin{abstract}
Using a combinatorial approach, we introduce the translated Whitney
numbers. This seems to be more natural than to write a product of
anarithmetical progression in terms of a power variable and
conversely.  We also extend our ideas to translated $r$-Whitney
numbers of both kinds and to translated Whitney-Lah numbers.
\end{abstract}

\section{Introduction}

It is well known that the Whitney numbers of the first and second kind are
defined, respectively, by the following recurrence relations%
\begin{eqnarray}
w_{\alpha }(n,k) &amp;=&amp;w_{\alpha }(n-1,k-1)+\left( \alpha (n-1)+1\right)
w_{\alpha }(n-1,k),\medskip  \label{w1} \\
W_{\alpha }(n,k) &amp;=&amp;W_{\alpha }(n-1,k-1)+\left( \alpha k+1\right) W_{\alpha
}(n-1,k).  \label{w22}
\end{eqnarray}

Benoumhani \cite{ben}  established the most properties of these numbers.
The principal ones are
\begin{equation}
\left( x+1|\alpha \right) _{n}=\sum_{k=0}^{n}(-1)^{n+k}w_{\alpha }(n,k)x^{k},
\label{FW1}
\end{equation}

\begin{equation}
x^{n}=\sum_{k=0}^{n}\text{\ }W_{\alpha }(n,k)\left( x-1|-\alpha \right) _{k},
\label{FW2}
\end{equation}%
where $\left( x|\alpha \right) _{n}=x\left( x+\alpha \right) \cdots \left(
x+\alpha \left( n-1\right) \right) $ and $\left( x|\alpha \right) _{0}=1$.

Our aim is to give a combinatorial interpretation of the coefficient of a
translation of these two expressions, and we call them the translated
Whitney numbers.

To understand the combinatorial interpretations given below, we introduce the nature of the elements we will use. When a given element occurs in
the same part as another element, the two elements create a conflict
situation of domination. One of them dominates the other, and then the
dominated one can mutate into $\alpha $ ways to another color, for example.
Then we can state that in each part containing more than one element, all
the elements have $\alpha $ possibilities to mutate to another color, except
the globally dominant element. Without loss of generality, we can consider
that the elements are in an ordered situation of domination: the first one
dominates the second, and so on. The $n^{th}$ element is the globally dominated
one.

\section{Translated Whitney numbers: a combinatorial approach}

In this section, we introduce the translated Whitney numbers of both
kinds.

\begin{definition}
The translated Whitney numbers of first kind, denoted $\stirlingf{n}{k}%
^{(\alpha )}$, count the number of permutations of $n$ elements with $k$
cycles such that the elements of each cycle can mutate in $\alpha $ ways,
except the dominant one.
\end{definition}

\begin{theorem}
The translated Whitney numbers of first kind satisfy the following
recurrence relation:
\begin{equation}
\stirlingf{n}{k}^{(\alpha )}=\stirlingf{n-1}{k-1}^{(\alpha )}+\alpha (n-1)%
\stirlingf{n-1}{k}^{(\alpha )},  \label{a1}
\end{equation}%
where $\stirlingf{0}{k}^{(\alpha )}=\delta _{k,0}$ and $\stirlingf{n}{0}%
^{(\alpha )}=\delta _{n,0}$, where $\delta $ is the Kronecker symbol.
\end{theorem}

\begin{proof}
Let us focus on the last element $n$ and, as specified in the introduction, it
is the globally dominated element. If it constitutes a cycle, we just have to
form $\left( k-1\right) $ cycles from the remaining $\left( n-1\right) 
$\ elements such that the elements of each cycle can mutate in $\alpha $ ways
except the dominant one, which gives $\stirlingf{n-1}{k-1}^{(\alpha )}$
possibilities. Otherwise, if it belongs to one of the cycles, we consider all the
permutations of $\left( n-1\right) $ elements with $k$ cycles such that the
elements of each cycle can mutate in $\alpha $ ways except the dominant one,
and we have $\stirlingf{n-1}{k}^{(\alpha )}$ possibilities; then adding the $%
n^{th}$ element to one of the cycles with $\alpha $ possibilities of mutation,
and we have $\alpha (n-1)$ possibilities. This gives the proof.
\end{proof}

\begin{remark}
For $\alpha =1$, we recover the classical Stirling numbers of the first kind.
\end{remark}

The translated Whitney of the first kind appears as a special case of the
generalization of the Stirling numbers given by Hsu and Shiue \cite{Hsu}; in
fact $\stirlingf{n}{k}^{(\alpha )}=S(n,k;-\alpha ,0,0)$. 

According to relation (\ref{FW1}), the translated Whitney numbers of the first kind
satisfy the following result:

\begin{theorem}
For $n\geq 1$, we have%
\begin{equation}
(x|\alpha )_{n}=\sum_{k=0}^{n}\stirlingf{n}{k}^{(\alpha )}x^{k}.  \label{p1}
\end{equation}
\end{theorem}

\begin{proof}
The proof is given by induction for the general case in \cite{Hsu}.
\end{proof}

\begin{remark}
Comparing to (\ref{FW1}), we understand why we call these numbers ``the
translated Whitney numbers''.
\end{remark}

According to \cite[Corollary 3.4]{Cakic}, the translated Whitney numbers
of the first kind satisfy the following combinatorial identity involving the
binomial coefficients:

\begin{corollary}
\begin{equation*}
\stirlingf{n}{k}^{(\alpha )}=\sum_{\substack{ i_{1}+\cdots +i_{n-1}=n-k \\ %
i_{j}\in \{0,1\}}}\binom{\alpha }{i_{1}}\binom{2\alpha }{i_{2}}\cdots \binom{%
\left( n-1\right) \alpha }{i_{n-1}}.
\end{equation*}
\end{corollary}

Now, we introduce the translated Whitney of the second kind.

\begin{definition}
The translated Whitney numbers of the second kind, denoted $\stirlings
{n}{k}^{(\alpha )}$, count the partitions of the set $\{1,2,\ldots,n\}$ into $k$
subsets such that the elements of each subset can mutate in $\alpha $ ways, except
the dominant one.
\end{definition}

\begin{theorem}
The translated Whitney numbers of second kind satisfy the following
recurrence relation:
\begin{equation}
\stirlings {n}{k}^{(\alpha )}=\stirlings {n-1}{k-1}^{(\alpha )}+\alpha
k\stirlings {n-1}{k}^{(\alpha )},  \label{a2}
\end{equation}%
with $\stirlings {0}{k}^{(\alpha )}=\delta _{k,0}$ and $\stirlings
{n}{0}^{(\alpha )}=\delta _{n,0}$.
\end{theorem}

\begin{proof}
Let us focus on the last element $n$. If it constitutes a singleton, then it remains
to form $\left( k-1\right) $ subsets from the $\left( n-1\right) $ remaining
elements such that the elements of each subset can mutate in $\alpha $ ways
except the dominant one, and we have $\stirlings {n-1}{k-1}^{(\alpha )}$
ways to do that. Otherwise, we constitute $k$ subsets from the $\left( n-1\right) $
elements such that the elements of each subset can mutate in $\alpha $ ways
except the dominant one, and we have $\stirlings {n-1}{k}^{(\alpha )}$
possibilities; then adding the $n^{th}$ element to one of the $k$ subsets with $%
\alpha $ possibilities of mutation, we get $\alpha k\stirlings
{n-1}{k}^{(\alpha )}$ possibilities. This ends the proof.
\end{proof}

\begin{remark}
For $\alpha =1$, we recover the classical Stirling numbers of the second
kind.
\end{remark}

The translated Whitney numbers of the second kind appear as a special case of the
generalization of the Stirling numbers given by Hsu and Shiue \cite{Hsu}. In
fact $\stirlings {n}{k}^{(\alpha )}=S(n,k;0,\alpha ,0)$. 

According to relation (\ref{FW2}), the translated Whitney numbers of the second kind
satisfy the following result:

\begin{theorem}
For $n\geq 1$, we have%
\begin{equation}
x^{n}=\sum_{k=0}^{n}\stirlings {n}{k}^{(\alpha )}(x|-\alpha )_{k}.
\label{p2}
\end{equation}
\end{theorem}

\begin{proof}
See \cite{Hsu}.
\end{proof}

According to \cite[Corollary 3.4]{Cakic}, we have

\begin{corollary}
\begin{equation*}
\stirlings {n}{k}^{(\alpha )}=\sum_{\substack{ i_{1}+\cdots +i_{n-1}=n-k
\\ i_{j}\in \{0,1\}}}\binom{\alpha }{i_{1}}\binom{\alpha (2-i_{1})}{i_{2}}%
\cdots \binom{\alpha (\left( n-1\right) -i_{1}-\cdots -i_{n-2})}{i_{n-1}}.
\end{equation*}
\end{corollary}

\section{The translated Whitney-Lah numbers}

In this section, we introduce the translated Whitney-Lah numbers using the
same operation of counting on ordered lists.

\begin{definition}
The translated Whitney-Lah numbers, denoted $\lah {n}{k}^{(\alpha )}$,
count the number of ways to distribute the set $%
\{1,2,...,n\}$ into $k$ ordered lists such that the elements of each list
can mutate with $\alpha $ ways, except the dominant one.
\end{definition}

\begin{theorem}
We have the following recurence relation:
\begin{equation}
\lah {n}{k}^{(\alpha )}=\lah
{n-1}{k-1}^{(\alpha )}+\alpha (n+k-1)\lah
{n-1}{k}^{(\alpha )}.  \label{a3}
\end{equation}
\end{theorem}

\begin{proof}
Let us consider the last element $n$. If it constitutes a list, we have to order the
remaining $\left( n-1\right) $ elements into $k$ lists such that the
elements of each list mutate with $\alpha $ ways except the dominant one, and
we have $\lah {n-1}{k-1}^{(\alpha )}$ ways to do that. Otherwise,
if $n$ belongs to one of $k$ constituted lists,
we have $\lah {n-1}{k}^{(\alpha )}$ possibilities.
So, we insert the $n^{th}$ element in one of the lists with $(n+k-1)$ 
choices ($\left( n-1\right) $
ways of putting the element after each ordered element and $k$ ways as a head list).
The $\alpha $ possibilities of
mutation gives us
the total number of $\alpha (n+k-1)\lah {n-1}{k}^{(\alpha )}$ possibilities.
\end{proof}

\begin{corollary}
We have%
\begin{equation}
\lah {n}{k}^{(\alpha )}=\sum_{j=k}^{n}\stirlingf{n}{j}%
^{(\alpha )}\stirlings {j}{k}^{(\alpha )}.  \label{cc}
\end{equation}
\end{corollary}
\begin{proof}
From (\ref{p1}) and (\ref{p2}), we can deduce our result.
\end{proof}

The translated Whitney-Lah numbers appear as a special case of the generalization
of the Stirling numbers given by Hsu and Shiue \cite{Hsu}. In fact $%
\lah {n}{k}^{(\alpha )}=S(n,k;-\alpha ,\alpha ,0)$.

According to relation (\ref{cc}), the translated Whitney-Lah numbers satisfy
the following result:

\begin{corollary}
\begin{equation}
(x|\alpha )_{n}=\sum_{k=0}^{n}\lah {n}{k}^{(\alpha
)}(x|-\alpha )_{k}.  \label{cb}
\end{equation}
\end{corollary}


\begin{remark}
For $\alpha =1$, we get the Lah numbers. See, for instance, \cite{hb}.
\end{remark}

According to \cite[Corollary 3.4]{Cakic}, we have

\begin{corollary}
\begin{equation*}
\lah {n}{k}^{(\alpha )}=\sum_{\substack{ i_{1}+\cdots
+i_{n-1}=n-k \\ i_{j}\in \{0,1\}}}\binom{2\alpha }{i_{1}}\binom{\alpha
(4-i_{1})}{i_{2}}\cdots \binom{\alpha (2\left( n-1\right) -i_{1}-\cdots
-i_{n-2}}{i_{n-1}}.
\end{equation*}
\end{corollary}

\section{The translated $r$-Whitney numbers}

Broder \cite{broder} gives a generalization of the Stirling numbers
of both kinds by adding a certain restriction the treatment of elements.
Using this generalization, Mezo \cite{Mez} treats the corresponding Bell
numbers: the $r$-Bell numbers. We give a corresponding extension of the translated
Whitney numbers of both kinds and the translated Whitney-Lah numbers in the sense of
Broder's approach.

\begin{definition}
The generalized translated Whitney numbers of the first kind (respectively
the second kind, Lah numbers), denoted $\stirlingf{n}{k}_{r}^{(\alpha )}$
(respectively $\stirlings {n}{k}_{r}^{(\alpha )}$, $\lah
{n}{k}_{r}^{(\alpha )}$), count the number of permutations (respectively
partitions, arrangements) of $n$ elements with $k$ cycles (respectively
parts, ordered lists) such that the $r$ first elements are in distinct
cycles (respectively parts, lists) and the elements of each cycle
(respectively parts, lists) can mutate in $\alpha $ ways, except the dominant
one.
\end{definition}

\begin{theorem}
The translated $r$-Whitney numbers satisfy the following recurrence relation:
\begin{eqnarray}
\stirlingf{n}{k}_{r}^{(\alpha )} &amp;=&amp;\stirlingf{n-1}{k-1}_{r}^{(\alpha
)}+\alpha (n-1)\stirlingf{n-1}{k}_{r}^{(\alpha )}, \\
\stirlings {n}{k}_{r}^{(\alpha )} &amp;=&amp;\stirlings {n-1}{k-1}_{r}^{(\alpha
)}+\alpha k\stirlings {n-1}{k}_{r}^{(\alpha )}, \\
\lah {n}{k}_{r}^{(\alpha )} &amp;=&amp;\lah
{n-1}{k-1}_{r}^{(\alpha )}+\alpha (n-1+k)\lah
{n-1}{k}_{r}^{(\alpha )},
\end{eqnarray}%
with $\stirlingf{0}{k}_{r}^{(\alpha )}=\stirlings {0}{k}_{r}^{(\alpha
)}=\lah {0}{k}_{r}^{(\alpha )}=\delta _{k,0}$ and $\stirlingf {n}{0}_{r}^{(\alpha )}=\stirlings {n}{0}_{r}^{(\alpha )}=\lah{n}{0}_{r}^{(\alpha )}=\delta _{n,0}$.
\end{theorem}

\begin{remark}
For $\alpha =1$, we obtain, respectively,
the $r$-Stirling numbers of the
first kind, the second kind and the $r$-Lah numbers.
\end{remark}

\begin{corollary}
We have
\begin{equation}
\lah {n}{k}_{r}^{(\alpha )}=\sum_{j=k}^{n}\stirlingf{n}{j}%
_{r}^{(\alpha )}\stirlings {j}{k}_{r}^{(\alpha )}.
\end{equation}
\end{corollary}

The translated $r$-Whitney of the first kind, the second kind and the $r$%
-Whitney-Lah numbers appear as special cases of the generalization of the
Stirling numbers given by Hsu and Shiue \cite{Hsu}. Indeed, $\stirlingf{n}{k}%
^{(\alpha )}=S(n,k;-\alpha ,0,r)$, $\stirlings {n}{k}_{r}^{(\alpha
)}=S(n,k;0,\alpha ,r)$ and $\lah {n}{k}_{r}^{(\alpha
)}=S(n,k;-\alpha ,\alpha ,r)$ and they satisfy the following results:

\begin{corollary}
For any nonnegative integers $0\leq r\leq k\leq n$, we have%
\begin{eqnarray}
(x+r|\alpha )_{n} &amp;=&amp;\sum_{k=0}^{n}\stirlingf{n}{k}_{r}^{(\alpha )}\left(
x+r\right) ^{k}, \\
(x+r)^{n} &amp;=&amp;\sum_{k=0}^{n}\stirlings {n}{k}_{r}^{(\alpha )}(x-r|-\alpha
)_{k}, \\
(x+r|\alpha )_{n} &amp;=&amp;\sum_{k=0}^{n}\lah
{n}{k}_{r}^{(\alpha )}(x-r|-\alpha )_{k}.
\end{eqnarray}
\end{corollary}

According to \cite[Corollary 3.4]{Cakic}, we have

\begin{corollary}
\begin{equation*}
\stirlingf{n}{k}_{r}^{(\alpha )}=\sum_{\substack{ i_{0}+\cdots +i_{n-1}=n-k \\ %
i_{j}\in \{0,1\}}}\binom{r}{i_{0}}\binom{r+\alpha }{i_{1}}\cdots \binom{%
r+\left( n-1\right) \alpha }{i_{n-1}},
\end{equation*}%
\begin{equation*}
\stirlings {n}{k}_{r}^{(\alpha )}=\sum_{\substack{ i_{0}+\cdots +i_{n-1}=n-k
\\ i_{j}\in \{0,1\}}}\binom{r}{i_{0}}\binom{r+\alpha (1-i_{0})}{i_{1}}\cdots 
\binom{r+\alpha (\left( n-1\right) -i_{0}-i_{1}-\cdots -i_{n-2})}{i_{n-1}},
\end{equation*}%
\begin{equation*}
\lah {n}{k}_{r}^{(\alpha )}=\sum_{\substack{ i_{0}+\cdots
+i_{n-1}=n-k \\ i_{j}\in \{0,1\}}}\binom{r}{i_{0}}\binom{r+(2-i_{0})\alpha }{%
i_{1}}\cdots \binom{r+\alpha (2\left( n-1\right) -i_{1}-\cdots -i_{n-2}}{%
i_{n-1}}.
\end{equation*}
\end{corollary}

\section{Acknowledgements}
The authors would like to thank the anonymous referee for careful
reading and suggestions that improved the clarity of this manuscript.

\begin{thebibliography}{99}
\bibliographystyle{plain}

\bibitem{hb} H. Belbachir and A. Belkhir, Cross recurrence relations for $r$%
-Lah numbers, {\it Ars Combinatoria}, \textbf{110} (2013), 199--203.

\bibitem{ben} M. Benoumhani, On Whitney numbers of Dowling lattices,
{\it Discrete Math.} \textbf{159} (1996), 13--33.

\bibitem{broder} A. Z. Broder, The $r$-Stirling numbers, {\it Discrete
Math.} \textbf{\ 49} (1984), 241--259.

\bibitem{Cakic} N. P. Cakic, B. S. El-Desouky, and G. V. Milovanovic,
Explicit formulas and combinatorial identities for generalized Stirling
numbers, {\it Mediterr. J. Math.} \textbf{10} (2013), 57--72.

\bibitem{knuth} R. L. Graham, D. E. Knuth, and O. Patashnik. {\it Concrete
Mathematics}, Addison-Wesley, 2nd edition, 1994.

\bibitem{Hsu} L. C. Hsu and P. J. Shiue, A unified approach to
generalized Stirling numbers, {\it Adv. in Appl. Math.} \textbf{20} (1998),
366--384.

\bibitem{Mez} I. Mezo, The $r$-Bell Numbers, {\it J. Integer Sequences}
\textbf{14} (2011), \href{https://cs.uwaterloo.ca/journals/JIS/VOL14/Mezo/mezo9.html}{Article 11.1.1}.


\end{thebibliography}


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\noindent 2000 {\it Mathematics Subject Classification}:  Primary 05A19; Secondary 11B73; 05A10.

\noindent {\it Keywords}: Stirling number, Whitney number, combinatorial identity,
combinatorial interpretation.

\bigskip
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\vspace*{+.1in}
\noindent
Received April 23 2013;
revised versions received  July 31 2013; September 10 2013.
Published in {\it Journal of Integer Sequences}, October 12 2013.

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