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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.
We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller th
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. Th
A {em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $mathbb{Z}timesmathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis
This paper completely characterizes the standard Young tableaux that can be reconstructed from their sets or multisets of $1$-minors. In particular, any standard Young tableau with at least $5$ entries can be reconstructed from its set of $1$-minors.
We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,ldots,V_n subset R^d$ be $n$ disjoint sets of points (of $n$ `colors). Suppose that for every $V_i$ and every point $v in V_i$ there are at