This work is concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebra associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are \emph{conveniently} interpreted as chains in the Young-Fibonacci lattice. A property of Killpatrick's evacuation is given a simpler proof, but this evacuation is no longer needed in making Roby's and Fomin's constructions coincide. We provide the set of Young-Fibonacci tableaux of size $n$ with a structure of graded poset, induced by the weak order on permutations of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients in the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okada's algebra. We end with a quite similar observation for four posets on Young-tableaux studied by Taskin.
The Andrews-Olsson identity and Bessenrodt insertion algorithm on Young walls
我们将Andrews-Olsson恒等式推广到双色分区。将量子仿射代数$\g_n=A^{(2)}_{2n}$，$A^{(2)}_{2n-1}$，$B^{(1)}_{n}$，$D^{(1)}_{n}$和$D^{(2)}_{n+1}$的真Young壁集作为双色划分集，推广的Andrews-Olsson恒等式意味着约化Young壁集的生成函数具有非常简单的公式：\begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{κ_i}$ where $κ_i=0$, $1$ or $2$, and $κ_i$ varies periodically. \end{center}此外，我们推广了Bessenrodt的算法，以另一种方式证明了推广的Andrews-Olsson恒等式.从这些算法中，我们可以给出严格划分对的某些子集上的晶体结构，这些严格划分对同构于$U_q(\g_n)$上的最高权模$V(Λ)$的晶体基$B(Λ)$。
String of columns rewriting and confluence of the jeu de taquin
Schützenberger的jeu de taquin算法是一种关于tableaux结构的算法，它通过tableaux列上的局部变换规则将倾斜tableaux变换为Young tableaux。该算法定义了相容于plactic同余的tableaux上的一个等价关系，并给出了Schur多项式的Littlewood-Richardson规则的一个证明.在这篇文章中，我们引入了列串重写系统的概念，作为列的胶合序列的转换机制。我们描述的jeudetaquin算法的执行重写路径的列重写的字符串。我们推导出代数性质的塑料同余式，我们涉及的jeu德taquin插入算法的tableaux。
The recording tableaux in the quantum Littlewood-Richardson map, the orthogonal transpose symmetry map, and the computation of $\mathfrak{k}$-highest weight tableaux
University of Coimbra（科英布拉大学）
本文组合证明了 Watanabe 给出的量子 LR 映射的满射性，并利用其逆映射计算和刻画了一类 $\mathfrak{k}$-最高权半标准杨表。
