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# From Dyck paths to standard Young tableaux

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### Abstract

The number of Dyck paths of semilength $$n$$ is certainly not equal to the number of standard Young tableaux (SYT) with $$n$$ boxes. We investigate several ways to add structure or restrict these sets so as to obtain equinumerous sets. Our most sophisticated bijective proof starts with Dyck paths whose $$k$$-ascents for $$k>1$$ are labeled by connected matchings on $$[k]$$ and arrives at SYT with at most $$2k-1$$ rows. Along the way, this bijection visits $$k$$-noncrossing and $$k$$-nonnesting partial matchings, oscillating tableaux and involutions with decreasing subsequences of length at most $$2k-1$$. In addition, we present bijections from eight other types of Dyck and Motzkin paths to certain classes of SYT.

### Author and article information

###### Journal
01 August 2017
1708.00513