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.
For a real number $t$, let $r_ell(t)$ be the total weight of all $t$-large Schr{o}der paths of length $ell$, and $s_ell(t)$ be the total weight of all $t$-small Schr{o}der paths of length $ell$. For constants $alpha, beta$, in this article we derive recurrence formulae for the determinats of the Hankel matrices $det_{1le i,jle n} (alpha r_{i+j-2}(t) +beta r_{i+j-1}(t))$, $det_{1le i,jle n} (alpha r_{i+j-1}(t) +beta r_{i+j}(t))$, $det_{1le i,jle n} (alpha s_{i+j-2}(t) +beta s_{i+j-1}(t))$, and $det_{1le i,jle n} (alpha s_{i+j-1}(t) +beta s_{i+j}(t))$ combinatorially via suitable lattice path models.
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 statistic `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.
Let $mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the skew three-rowed strip $mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were derived automatically and proved rigorously by Zeilberger via his powerful program and WZ method. It appears that each one is a linear combination of Motzkin numbers with constant coefficients. In this paper we will introduce a simple bijection between Motzkin paths and standard Young tableaux with at most three rows. With this bijection we answer Zeilbergers question affirmatively that there is a uniform way to construct bijective proofs for all of those identites.
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
