Combinatorics of Partial Derivatives

Short formulas for algebraic covariant derivative curvature tensors via Algebraic Combinatorics

We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if…

部分解析与非交叉组合学

Partial Resolutions and Noncrossing Combinatorics

本文研究了有限可约群的Hecke代数中心元素，通过部分Springer解析和非交叉集合构造，扩展了相关不变量的计算方法。

Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is mi…

Schur Partial Derivative Operators

A lattice diagram is a finite list L=((p_1,q_1),...,(p_n,q_n) of lattice cells. The corresponding lattice diagram determinant is Δ_L(X;Y)=\det \| x_i^{p_j}y_i^{q_j} \|. These lattice diagram determinants are crucial in the study of the so-called ``n! conjecture'' of A. Garsia and M. Haiman. The space M_L is the space spanned by all partial derivatives of Δ_L(X;Y). The ``shift operators'', which are particular partial symmetric derivative operators are very useful in the comprehension of the structure of the M_L spaces. We describe here how a Schur function partial derivative operator acts on l…

The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities

We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.

斜舒尔多项式的对角导数

The diagonal derivative of a skew Schur polynomial

本文给出斜舒尔多项式在对角微分算子作用下的显式公式，推广了 Weigandt 对普通舒尔多项式的结果。

Borel偏序集的Borel链条件

Borel chain conditions of Borel posets

本文在描述性组合学中研究Borel偏序集的链条件分类，证明了存在与经典情形类似但不同于其他Borel组合学对应物的链条件层级。

Analysis and combinatorics of partition zeta functions

We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions…

Combinatorics of rational functions and Poincare-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form

We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie albegras A_r, B_r, C_r, D_r in a conviniently chosen Poincare-Birkhoff-Witt basis. As a byproduct we obtain results on the combinatorics of rational functions, namely non-trivial identities are proved between certain rational functions with partial symmetries.

Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\HH(b)$, where $b$ is in the unit ball of $H^\infty(\CC_+)$. In particular, we generalize a result of Ahern--Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{ω,n}^b$ of the evaluation of $n$-th derivative of elements of $\HH(b)$ a…

部分Theta函数的首根的组合学

The Combinatorics of the Leading Root of the Partial Theta Function

本文通过有根树与堆叠多联骨牌或Ferrers图的组合解释，给出了部分Theta函数首根系数的显式组合意义，并推导了其渐近公式。

Five interpretations of Faà di Bruno's formula

In these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads.