Reduction of $m$-Regular Noncrossing Partitions

Details Reduction of $m$-Regular Noncrossing Partitions Reduction of $m$-Regular Noncrossing Partitions

2004-06-09

arxiv.org/abs/math/0406180v1

Mathematics

Combinatorics

Scientific paper

In this paper, we present a reduction algorithm which transforms $m$-regular partitions of $[n]=\{1, 2, ..., n\}$ to $(m-1)$-regular partitions of $[n-1]$. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers.

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