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{\bf Shu-Chung Liu, Yi Wang and Yeong-Nan Yeh}
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{\bf Chung-Feller Property in View of Generating Functions}
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The classical Chung-Feller Theorem offers an elegant perspective for
enumerating the Catalan number $c_n= \frac{1}{n+1}\binom{2n}{n}$. One
of the various proofs is by the uniform-partition method. The method
shows that the set of the free Dyck $n$-paths, which have
$\binom{2n}{n}$ in total, is uniformly partitioned into $n+1$ blocks,
and the ordinary Dyck $n$-paths form one of these blocks; therefore
the cardinality of each block is $\frac{1}{n+1}\binom{2n}{n}$. In this
article, we study the Chung-Feller property: a sup-structure set can
be uniformly partitioned such that one of the partition blocks is
(isomorphic to) a well-known structure set. The previous works about
the uniform-partition method used bijections, but here we apply
generating functions as a new approach. By claiming a functional
equation involving the generating functions of sup- and sub-structure
sets, we re-prove two known results about Chung-Feller property, and
explore several new examples including the ones for the large and the
little Schr\"{o}der paths. Especially for the Schr\"{o}der paths, we
are led by the new approach straightforwardly to consider ``weighted''
free Schr\"{o}der paths as sup-structures. The weighted structures are
not obvious via bijections or other methods.
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