Bijective Proofs for "Enumerative Properties of Ferrers Graphs"

Enumerative properties of Ferrers graphs

We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.

A bijective proof of the hook-length formula for shifted standard tableaux

We present a bijective proof of the hook-length formula for shifted standard tableaux of a fixed shape based on a modified jeu de taquin and the ideas of the bijective proof of the hook-length formula for ordinary standard tableaux by Novelli, Pak and Stoyanovskii. In their proof Novelli, Pak and Stoyanovskii define a bijection between arbitrary fillings of the Ferrers diagram with the integers $1,2,...,n$ and pairs of standard tableaux and hook tabloids. In our shifted version of their algorithm the map from the set of arbitrary fillings of the shifted Ferrers diagram onto the set of shifted …

Enumerative and bijective aspects of combinatorial maps: generalization, unification and application (PhD thesis)

This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumer…

EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model

A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau. We show that the set of EW-tableaux on a given Ferrers diagram are i…

代数与枚举组合学中的简短证明

Short Proofs in Algebraic and Enumerative Combinatorics

本文利用ChatGPT 5.4 Pro自主生成简短证明，解决了模格上的梯队运动算子、停车函数统计以及plactic幺半群中心化子等领域的若干开放问题。

Bijections for inversion sequences, ascent sequences and 3-nonnesting set partitions

Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's algorithm, Lin confirmed a conjecture due independently to the author and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-nonnesting partitions have the same cardinality. In this paper, we provide a bijective proof of this conjecture. Our bijection also enables us to provide a new bijective proof of …