From Dyck paths to standard Young tableaux

Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner

Published 2017-08-01Version 1

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.