Graph insertion operads

Binary Search Tree insertion, the Hypoplactic insertion, and Dual Graded Graphs

Fomin (1994) introduced a notion of duality between two graded graphs on the same set of vertices. He also introduced a generalization to dual graded graphs of the classical Robinson-Schensted-Knuth algorithm. We show how Fomin's approach applies to the binary search tree insertion algorithm also known as sylvester insertion, and to the hypoplactic insertion algorithm.

A proof of Pieri's formula using the generalized Schensted insertion algorithm for rc-graphs

We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.

基于子图收缩的算子代数的一种推广

A generalization of operads based on subgraph contractions

本文提出算子代数的推广——收缩代数，其操作集由连通图索引、复合规则由连通子图收缩编号，证明了多种经典算子代数可推广至该框架，并验证了Koszul对偶与Gröbner基等工具的适用性及特定收缩代数的Koszul性质。

$n\text{-}Lie_d$ 算子及其 Koszul 对偶

$n\text{-}Lie_d$ Operad and its Koszul Dual

本文研究 $n\text{-}Lie_d$ 算子及其 Koszul 对偶 $n\text{-}Com_{-d+n-2}$，通过 Specht 模和奇图谱揭示两者联系。

曲面中的环、弦图、交错图：算子分解与生成文法

Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars

本文研究闭曲面中 filoop 的亏格，通过弦图和交错图计算最小亏格，给出球和与环和的典范分解，并构造生成文法，建立稳定性性质。

Finitely Dependent Insertion Processes

A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance greater than $k$ receive independent colorings. Holroyd and Liggett constructed the first stationary $k$-dependent $q$-colorings by introducing an insertion algorithm on the complete graph $K_q$. We extend their construction from complete graphs to weighted directed graphs. We show that complete multipartite analogues of $K_3$ and $K_4$ are the only graphs who…

Generalization of Schensted insertion algorithm to the cases of hooks and semi-shuffles

Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above.

Gelfand $S_n$-图的插入算法

Insertion algorithms for Gelfand $S_n$-graphs

本文改进 Beissinger 的受限 RS 插入，提出在列末添加单元格的新双射，并利用两种插入算法分类一般对称群完美模型对应 $W$-图中的分子及猜想的胞腔。

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here includ…

Dual graded graphs for Kac-Moody algebras

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(Γ_s,Γ_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The graded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeled versions of the strong and weak orders of $W$ respectively. Using a construction of Lusztig for quivers with an admissible automorphism, we define folded insertion for a Kac-Moody algebra and obtain Sagan-Worley shifted insertion from Robinson-Schensted insertion as a special case. Drawing on work of Stembridge, we analyze the induced subgraphs of $(Γ_s,Γ_w)$ which are distri…

RC-graphs and a generalized Littlewood-Richardson rule

Using a generalization of the Schensted insertion algorithm to rc-graphs, we provide a Littlewood-Richardson rule for multiplying certain Schubert polynomials by Schur polynomials.

An effective crossing minimisation heuristic based on star insertion

We present a new heuristic method for minimising crossings in a graph. The method is based upon repeatedly solving the so-called {\em star insertion problem} in the setting where the combinatorial embedding is fixed, and has several desirable characteristics for practical use. We introduce the method, discuss some aspects of algorithm design for our implementation, and provide some experimental results. The results indicate that our method compares well to existing methods, and also that it is suitable for dense instances.

Edge partitions of the complete graph and a determinant like function

In this paper we prove the case $dim(V_3)=3$ of a conjecture about the exterior operad $Λ^{S^2}_{V_d}$. For this we introduce a collection of natural involutions on the set of homogeneous cycle-free $d$-partitions of the complete graph $K_{2d}$, and show that these involutions correspond to the relations in $Λ^{S^2}_{V_d}(2d+1)$. When $d=3$ this correspondence allows us to give an explicit description of a determinant-like map and to settle the above mentioned conjecture.

Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical v…