Standard Young Tableaux and Colored Motzkin Paths

In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the $n$-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length $n$. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most $2d+1$ rows and the set of SYTs with at most 2d rows.Comment: 21 page

On qq-Counting of Noncrossing Chains and Parking Functions

For a finite Coxeter group $W$, Josuat-Verg\`es derived a $q$-polynomial counting the maximal chains in the lattice of noncrossing partitions of $W$ by weighting some of the covering relations, which we call bad edges, in these chains with a parameter $q$. We study the connection of these weighted chains with parking functions of type $A$ ($B$, respectively) from the perspective of the $q$-polynomial. The $q$-polynomial turns out to be the generating function for parking functions (of either type) with respect to the number of cars that do not park at their preferred spaces. In either case, we present a bijective result that carries bad edges to unlucky cars while preserving their relative order. Using this, we give an interpretation of the $\gamma$-positivity of the $q$-polynomial in the case that $W$ is the hyperoctahedral group.Comment: 32 pages, to be published in SIDM