Olexandr Ganyushkin, Volodymyr Mazorchuk
We obtain several combinatorial results about chains, cycles and orbits of the elements of the symmetric inverse semigroup $\IS_n$ and the set $T_n$ of nilpotent elements in $\IS_n$. We also get some estimates for the growth of $|\IS_n|$ and $|T_n|$, and study random products of elements from $\IS_n$.
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We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n, \text{diag}(S_k) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves less products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.
We study some combinatorial properties of partial wreath $k$-th power of the semigroup $\mathcal{IS}_d$. In particular, we calculate its order, the number of idempotents and the number of D-classes.
We investigate the combinatorics of the general formulas for the powers of the operator $h \partial^k$, where $h$ is a central element of a ring and $\partial$ is a differential operator. This generalizes previous work on the powers of operators $h \partial$. New formulas for the generalized Stirling numbers are obtained.
The natural forms of the Leibniz rule for the $k$th derivative of a product and of Faà di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if a…
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。
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…
We describe two new combinatorial algorithms (using the language of "triangular arrays") for computing the Fourier transforms of simple perverse sheaves on the moduli space of representations of an equioriented quiver of type A. (A rather different solution to this problem was previously obtained by Knight-Zelevinsky.) Along the way, we also show that the closure partial order and the dimensions of orbits have especially concise descriptions in the language of triangular arrays.
We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.
We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generati…
Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $σ$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $Ω_σ\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of ro…
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。
Fractional Helly property and combinatorics of forking in NTP$_2$ theories
本文研究了满足分数Helly性质的理论，这些理论扩展了NIP理论，并形成了一种新的低NTP₂理论子类。文章给出了许多新例子，并探讨了分叉和f-通用性在可计算群中的性质，同时提出了关于NTP₂理论中分叉有限组合性质的猜想。
In this article, we investigate the image and preimage of the important combinatorial sets such as central sets, $C$-sets, and $J_δ$-sets which play an important role in the study of combinatorics under certain partial semigroup homomorphism. Using that we prove certain results which deal with the existence of $C$-set which are not central in partial semigroup framework.
Orbits and equivariant local systems combinatorics in graded Lie algebras
本文用组合方法刻画四类无限族分次李代数（辛群与特殊正交群各两类）中的轨道、Jordan 块分解、维数、偏序及等变局部系统，为研究分次李代数产生的反常层提供工具。
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。
We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.
On Geometry and Combinatorics of Finite Classical Polar Spaces
本文研究有限经典极空间的几何与组合特性，探讨了spread、ovoid及正则系统的存在性与界等问题，并阐述其与正则图、区组设计等组合对象的关联。
We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the underlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are th…
We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full …
Toric degenerations of partial flag varieties and combinatorial mutations of matching field polytopes
本文通过匹配场多面体和组合突变，构造了部分旗簇的新环面退化，并证明其正规性及与Newton-Okounkov体的联系，最终应用于小规模格拉斯曼流形。
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced 'double posets', that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the 'double order polytope' and the 'double chain polytope' and we …
On unirational varieties with poset parameterizations
研究利用偏序集为特定有理映射下像的扎里斯基闭包提供参数化，将代数几何等领域中多个问题转化为组合学问题，通过各领域实例展示该技术的应用。
Compact-form solution to the time-dependent Schrödinger equation with an arbitrary potential
本研究结合偏微分方程、计数组合学与数论方法，得到任意维度任意势下抛物型偏微分方程的紧致形式精确解，探讨了其在多领域的优势与前景。
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