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 interpre
tation 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.
Let $C_n$ be the set of Dyck paths of length $n$. In this paper, by a new automorphism of ordered trees, we prove that the statistic `number of exterior pairs, introduced by A. Denise and R. Simion, on the set $C_n$ is equidistributed with the statis
tic `number of up steps at height $h$ with $hequiv 0$ (mod 3). Moreover, for $mge 3$, we prove that the two statistics `number of up steps at height $h$ with $hequiv 0$ (mod $m$) and `number of up steps at height $h$ with $hequiv m-1$ (mod $m$) on the set $C_n$ are `almost equidistributed. Both results are proved combinatorially.
A permutation $sigmainmathfrak{S}_n$ is simsun if for all $k$, the subword of $sigma$ restricted to ${1,...,k}$ does not have three consecutive decreasing elements. The permutation $sigma$ is double simsun if both $sigma$ and $sigma^{-1}$ are simsun.
In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.
