Schröder Paths and Pattern Avoiding Partitions

Details Schröder Paths and Pattern Avoiding Partitions Schröder Paths and Pattern Avoiding Partitions

2008-05-16

arxiv.org/abs/0805.2465v2

Mathematics

Combinatorics

6 pages

Scientific paper

In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined enumeration of 12312-avoiding (resp. 12321-avoiding) partitions according to the number of blocks can be reduced to the enumeration of certain Schr\"oder paths according to the number of peaks. Furthermore, we get the enumeration of irreducible 12312-avoiding (resp. 12321-avoiding) partitions, which are closely related to skew Dyck paths.

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