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\auteurcourant={B. VOIGT}
\titrecourant={COMMON GENERALIZATION OF COEFFICIENTS}
\pageno=87

\eightpoint
\leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1984$, 229/S--08}
\leftline{Actes 8\emini\ S\'eminaire Lotharingien, p. 87--89}

\tenpoint
\let\it=\sl

\vglue 1.5cm
\centerline{\bf A COMMON GENERALIZATION OF BINOMIAL}
\vskip 5pt
\centerline{\bf COEFFICIENTS, STIRLING NUMBERS AND}
\vskip 5pt
\centerline{\bf GAUSSIAN COEFFICIENTS}
\vskip 2.8mm
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{{\petcap Bernd} VOIGT}
\vskip 5mm
Let $A_0$, $A_1$, $A_2$, \dots\ be finite sets. For nonnegative
integers $n$ and $k$ denote by
$S_k^n(a_0,a_1,a_2,\dots\,)$,
where $a_i=\left|A_i\right|$,
the number of words $w=(w_0,\ldots,w_{n-1})$ such that

\vskip 5pt
(1) $w$ contains $k$ labels, say at positions
$i_0,\ldots,i_{k-1}$,

\vskip 5pt
(2) all entries in $w$ before position $i_0$ belong to $A_0$,
all entries in $w$ between positions $i_l$ and $i_{l+1}$,
where $l=0,\ldots, k-2$, belong to $A_{l+1}$,
all entries after position $i_{k-1}$ belong to $A_k$.

\midinsert
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\multispan4$i_{k-1}$\hfil\cr}}$$
\endinsert

As $\displaystyle S_n^k(\vec a)
=\sum _{0\le i_0<i_1<\cdots<i_{k-1}<n}
a_0^{i_0}\cdot
a_1^{i_1-i_0-1}
\cdots
a_k^{n-i_{k-1}-1}$,
the numbers $S_k^n$ can be obviously defined for sequences of
complex numbers.

\vskip 10pt
\noindent 
{\it  Examples}.

\nobreak
\vskip 5pt
\halign{\indent #&\enspace$#$\hfil&\qquad
\vtop{\hsize 6cm \noindent
\strut #\strut}\cr
(1)&S_n^k(1,1,\ldots\,)={n\choose k}&(Binomial coefficients)\cr
\noalign{\vskip 5pt}
(2)&S_k^n(0,1,2,\ldots\,)=S_k^n&(Stirling numbers of the 
second kind)\cr 
\noalign{\vskip 5pt}
(3)&S_k^n(1,q,q^2,\ldots\,)={n\choose k}_q&(Gaussian 
binomial coefficients)\cr
\noalign{\vskip 5pt}
(4)&S_k^n(q,q^2,q^3,\ldots\,)={}&number of affine 
$k$-dimensional
subspaces in the $n$-dimensional affine space over $GF(q)$.\cr
\noalign{\vskip 5pt}
(5)&S_k^n(2,3,4,\ldots\,)={}&number of Boolean sublattices 
$P(k)$ in $P(n)$ ($P(n)\equiv {}$lattice of subsets of 
an $m$-element set).\cr}

\vskip 10pt
\noindent
{\it Pascal-identity} : $S_k^{n+1}(\vec a)
=S_{k-1}^n+a_k\cdot S_k^n$.

\vskip 5pt \noindent
{\it explicitly} : $\displaystyle S_k^n(\vec a)
=\sum _{i=0}^k a_i^n\cdot
\prod _{j=0\;j\not=i}(a_i-a_j)^{-1}$,
provided $a_0,\ldots,a_k$ are mutually distinct.

\vskip 5pt
Let $P_0^{\vec a}(x)=1$ and 
$P_{k+1}^{\vec a}(x)=(x-a_k)\cdot P_k^{\vec a}(x)$,
i.e. 
  $$P_k^{\vec a}(x)=(x-a_{k-1})\cdots (x-a_0).$$
{\it Inversion, resp. interpolation formulae} :
  $$x^n=\sum _{k\ge 0} S_k^n(\vec a)\cdot P_k^{\vec a}(x).$$
Now one can introduce the inverse numbers $s_k^n(\vec a)$ by
$\sum _js_j^n(\vec a)\cdot S_k^j(\vec a)=\delta _k^n$ (Kronecker delta).

\vskip 5pt
\noindent 
{\it Recursion} : $s_k^{n+1}(\vec a)=s_{k-1}^n - a_n\cdot
s_k^n$.

\vskip 5pt
\noindent 
These numbers can be used in order to describe inversion for
arbitrary ascending sequences of normalized polynomials :
let 
  $$\displaylines{
S_k^n(\vec a,\vec b)=\sum _j s_j^n(\vec a)\cdot 
S_k^j(\vec b).\cr
\noalign{\hbox{then}}
P_n^{\vec a}(\vec x)
=\sum_k S_k^n(\vec a,\vec v)\cdot P_k^{\vec b}(x).\cr}
$$
{\it Recursion} :
$S_k^{n+1}(\vec a,\vec b)
=S_{k-1}^n (\vec a,\vec v)+(b_k-a_k)\cdot 
S_k^n (\vec a,\vec b)$.

\medskip
\th Theorem|Let $\vec a$ as before, $l$ a complex number, then

\noindent
\decale{\rm (i)}|$S_k^n(\vec a)=\displaystyle 
\sum_j{n\choose j}\cdot l^{n-j}\cdot S_k^j(\vec a-l)$,
\decale{\rm (ii)}|$S_k^n(\vec a)=\displaystyle 
\sum_j{j\choose k}\cdot l^{j-k}\cdot S_j^n(\vec a-l)$,
\decale{\rm (iii)}|$s_k^n(\vec a)
=\sum _j{n\choose j}\cdot l^{n-j}\cdot 
s_k^j(\vec a+l)$,
\decale{\rm (iv)}|$s_k^n(\vec a)
=\sum _j{j\choose k}\cdot l^{j-k}\cdot 
s_j^n(\vec a + l)$.
\finth


\rem Remark|The case $\vec a=(1,q,q^2,\ldots\,)$ and
$l=1$ is a theorem of \pc CARLITZ|~[1].

\th Corollary|The number of Boolean sublattices of
$P(n)$ is equal to the $(n+2)$-nd Bell number $B_{n+2}$.
\finth

\vskip 1cm
\goodbreak
\eightpoint
\centerline{REFERENCES}
\nobreak
\vskip .5cm
\article 1|\pd CARLITZ (L.)|On abelian fields|Trans.
Amer. Math. Soc.|35|1933|122--126|

\tenpoint

\adresse
Bernd {\petcap Voigt},
Fakult\"at f\"ur Mathematik
der Universit\"at,
Postfach 8640,
D-48 Bielefeld.

\fin

\bye
