KAIST Discrete Math Seminar

List of speakers

• Tuesday 10:30AM-12PM Seunghyun Seo (서승현), Kangwon National University, Refined enumeration of trees by the size of maximal decreasing trees
• Tuesday 1:30PM-3PM HwanChul Yoo (유환철), KIAS, Specht modules of general diagrams and their Hecke counterparts
• Tuesday 4PM-5:30PM Heesung Shin (신희성), Inha University, q-Hermite 다항식을 포함하는 두 항등식에 관하여
• Wednesday 10:30AM-12PM Soojin Cho (조수진), Ajou University, Skew Schur P-functions
• Wednesday 1:30PM-3PM Sangwook Kim (김상욱), Chonnam National University, Flag vectors of polytopes

Laguerre histories are certain colored Motzkin paths with some weight for each elementary steps. In this talk, we study two famous bijections between permutations and Laguerre histories, made by Francon-Viennot and Foata-Zeilberger. This two bijections are enable us to give permutations as combinatorial interpretations of continued fractions. The former is associated in linear statistics and the latter be in cyclic statistics. Using two mappings, we are able to make various results about several statistics of permutations.

For a labeled tree on the vertex set [n]:={1,2,…,n}, define the direction of each edge ij as i to j if i<j. The indegree sequence λ=1^e₁2^e₂ … is then a partition of n-1. Let a_λ be the number of trees on [n] with indegree sequence λ. In a recent paper (arXiv:0706.2049v2) Cotterill stumbled across the following remarkable formulas  where k = Σ_i e_i. In this talk, we first construct a bijection from (unrooted) trees to rooted trees which preserves the indegree sequence. As a consequence, we obtain a bijective proof of the formula. This is a joint work with Jiang Zeng.