## Parking functions and noncrossing partitions.(English)Zbl 0883.06001

Electron. J. Comb. 4, No. 2, Research paper R20, 14 p. (1997); printed version J. Comb. 4, No. 2, 261-274 (1997).
Summary: A 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 $$a_1,\dots, a_n$$, then $$b_i\leq i$$. A 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 $$[n+1]$$ 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 $${\mathfrak S}_n$$ on NC$$_{n+1}$$.

### MSC:

 06A07 Combinatorics of partially ordered sets 05E05 Symmetric functions and generalizations 05A15 Exact enumeration problems, generating functions 05E10 Combinatorial aspects of representation theory 05E25 Group actions on posets, etc. (MSC2000)
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