Simion猜想和对数凹性

Log-Concavity and Simion's Conjecture

以N(m,n;λ，μ）表示在m×n的矩形格的左上角和右下角分别删掉分拆λ和μ的Ferrers图后从左下角到右上角格路的数目。Simion猜想对任意分拆λ，N（l-k,k;λ,φ)关于k是对数凹的，本文证明了，如果序列x0,x1,…，xn为对数凹的，则序列yk=∑i=k^n(a+i b+k)xi亦为对数凹的，并给出其对Simion猜想的应用。本文还证明对所有分拆λ和μ，N（l-k,k;λ，μ）关于k是对数凹的。

Denote by N(m, n;λ,μ) the number of lattice paths from the lower left corner to the upper right corner in an m×n grid with the Ferrers diagrams of two partitions λ and μ being removed from the upper left corner and the lower right corner respectively. Simion's conjectured that N(e-k, k;λ,θ) is log-concave in k. Here we show that if the sequence x_0, x_1,..., x_n is log-concave, then the sequence y_k x_i is also log- concave, and present an application of this result to Simion's conjecture. Furthermore, we show that N(e-k, k;λ,μ) is log-concave in k for all partitions λand μ