No Arabic abstract
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 least $delta |V_i|$ other points $u in V_i$ so that the line connecting $v$ and $u$ contains a third point of another color. Then the union of the points in all $n$ sets is contained in a subspace of dimension bounded by a function of $n$ and $delta$ alone.
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 study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Millikens theorem on trees.
We define an excedance number for the multi-colored permutation group, i.e. the wreath product of Z_{r_1} x ... x Z_{r_k} with S_n, and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(pi) and fix(pi) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fixed number of excedances and absolute fixed points.
Let $Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $varepsilon>0$ the asymptotic formula $$ sum_{nle x} LambdaBig(Big[frac{x}{n}Big]Big) = xsum_{dge 1} frac{Lambda(d)}{d(d+1)} + O_{varepsilon}big(x^{9/19+varepsilon}big) qquad (xtoinfty)$$ holds. This improves a recent result of Bordell`es, which requires $frac{97}{203}$ in place of $frac{9}{19}$.
Motivated by Alladis recent multi-dimensional generalization of Sylvesters classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchys identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.