Schur Polynomials through Lindström Gessel Viennot Lemma

为使用 Gessel-Viennot 行列式合理表达带旗 Schur 函数所需的行界

Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinants

本文推广带旗 Schur 函数，去掉 β 弱递增限制，给出 (λ, β) 非置换的充要条件，并分类有效行界，其计数为抛物 Catalan 数，且可用于表达 Demazure 特征。

Combinatorial Interpretations of the q-Faulhaber and q-Salie Coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salie's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.

Viewing determinants as nonintersecting lattice paths yields classical determinantal identities bijectively

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot interpretation of semistandard Young tableaux and the Jacobi-Trudi identity together with elementary observations. After some preparations, this point of view provides very simple "graphical proofs" for classical determinantal identities like the Cauchy--Binet formula, Dodgson's condensation formula, the Plücker relations and Laplace's expansion. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which m…

A planar network proof for Hankel total positivity of type $B$ Narayana polynomials

The Hankel matrix of type B Narayana polynomials was proved to be totally positive by Wang and Zhu, and independently by Sokal. Pan and Zeng raised the problem of giving a planar network proof of this result. In this paper, we present such a proof by constructing a planar network allowing negative weights, applying the Lindström-Gessel-Viennot lemma and establishing an involution on the set of nonintersecting families of directed paths.

A normalization formula for the Jack polynomials in superspace and an identity on partitions

We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.

不相交路径与树的距离矩阵的行列式

Non-intersecting paths and the determinant of the distance matrix of a tree

本文通过符号反转对合与Lindström-Gessel-Viennot引理，首次给出了树的距离矩阵行列式的Graham-Pollak公式的组合证明，并为理解其现有推广与$q$-模拟提供了统一框架。

An Introduction to Schur Polynomials

Notes from a course at the ATM Workshop on Schubert Varieties, held at The Institute of Mathematical Sciences, Chennai, in November 2017. Various expansions of Schur functions, the Lindström-Gessel-Viennot lemma, semistandard Young tableaux, Schensted's insertion algorithm, the plactic monoid, the RSK correspondence, and the Littlewood-Richardson rule are discussed.

卷积Catalan数的Hankel行列式与非相交格路径：Cigler猜想的一个双射证明

Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

本文利用反射原理、Lindström-Gessel-Viennot方法及非相交格路径的反射叠合构造，为Cigler关于卷积Catalan数Hankel行列式的猜想提供了一个双射证明。

Planar graphs and Stanley's Chromatic Functions

This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gesse…

Schur functions and alternating sums

We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations; notably we shall discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and its dual (Von N{ä}gelsbach-Kostka) identity, their proofs using the correspondence with lattice paths of Gessel and Viennot, and finally the Littlewood-Richardson rule. Our our goal is to show that the mentioned statements are closely related, and can be proved using variations of the same basic technique. We also want to emphasise the central part that is played by matrices o…

Wilson's theorem modulo p^2 derived from Faulhaber polynomials

First, we present a new proof of Glaisher's formula dating from 1900 and concerning Wilson's theorem modulo p^2. Our proof uses p-adic numbers and Faulhaber's formula for the sums of powers (17th century), as well as more recent results on Faulhaber's coefficients obtained by Gessel and Viennot. Second, by using our method, we find a simpler proof than Sun's proof regarding a formula for (p-1)! modulo p^3, and one that can be generalized to higher powers of p. Third, we can derive from our method a way to compute the Stirling numbers modulo p^3, thus improving Glaisher and Sun's own results fr…