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\noindent {\bf Richard P.\ Stanley }
\bigskip\noindent Parking Functions and Noncrossing Partitions
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A  {\sl parking function} is a sequence $(a_1,\dots,a_n)$ of
positive integers such that,\ if $b_1\leq b_2\leq \cdots\leq b_n$ 
is the increasing rearrangement of the sequence $(a_1,\dots, a_n),$
then $b_i\leq i$. A {\sl noncrossing partition} of the set 
$[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with
 the property that if $a<b<c<d$ and some block $B$ of $\pi$ contains
 both $a$ and $c$, while some block $B'$ of $\pi$ contains both 
$b$ and $d$, then $B=B'$. We establish some connections between 
parking functions and noncrossing partitions. A generating function 
for the flag $f$-vector of the lattice NC$_{n+1}$ of noncrossing
 partitions of $[{\scriptstyle n+1}]$ is shown to coincide (up to 
the involution $\omega$ on symmetric function) with Haiman's parking
 function symmetric function. We construct an edge labeling of
 NC$_{n+1}$ whose chain labels are the set of all parking functions 
of length $n$. This leads to a local action of the symmetric
group ${S}_n$ on NC$_{n+1}$.
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