Symmetric Joint Distributions on Relative Statistics

Abstract

對於一個 π ∈ Sn, 我們可以定義 σ ∈ Sn 相對於 π 的相對逆序 inv_π(σ) (relative inversion with respect to π), 這是與原來 π ∈ Sn 上的 inv 等分佈的統計量. 此定義推廣了 Gillespie 等人對 MacDonald 多項式的對稱性的研究中所定義的 k-inversion. 本篇論文的主要結果 (Theorem 2.2.3) 是證明若 π1,π2 兩個排列上在 weak Bruhat order 上有連線且恰為一組相鄰位置對調, 則這兩個統計量 inv_π1, inv_π2 所形成的聯合分佈 (joint distribution) 是對稱的. 亦即 (inv_π1, inv_π2) ∼ (inv_π2, inv_π1). 此外, 對於更細緻的聯合對稱分佈現象與 relative descent, relative major index 等, 我們也提出一些觀察與猜想.For a permutation π∈ Sn, we can define the relative inversion inv_π(σ) of σ ∈ Sn with respect to π. The statistic inv_π has the same distribution with the standard inversion statsitics inv over Sn. This definition is motivated and generalized the k-inversion defined by Gillespie et al. in their work of seeking a combinatorial proof of the (still open) famous symmetric property of the MacDonald polynomials. The main result of this thesis (Theorem 2.2.3) is to prove that if π1, π2 is connected in the weak Bruhat order, then the two statistics inv_π1 and inv_π2 have the symmetric joint distribution (inv_π1, inv_π2) ∼ (inv_π2, inv_π1). Further observations on symmetry, relative descent and relative major index are also given.