Generating Functions of Nestohedra and Applications
Andrew G. Fenn
我们通过生成函数研究了由完全二部图生成的嵌套面族，并证明了它们的一些组合不变量。
We examine the family of nestohedra resulting from the complete bipartite graph through the medium of a generating function and demonstrate some of their combinatorial invariants.
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For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $Σ_Q$ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf algebra morphism from a combinatorial Hopf algebra of building sets. For the class of graph-associahedra the corresponding quasisymmetric function is a new isomorphism invariant of graphs. The obtained invariant is quite natural as it is the generating function of ordered colorings of graphs and satisfies the recurrence relation with respect to deletions of vertice…
A Möbius function computing $h$-polynomials of nestohedra
本文引入加权超图偏序集，其 Möbius 值给出 nestohedron 的 $h$-多项式系数，并据此提出两个新递推关系。
A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function $F(P_B)$ of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron $P_B$ associated to a building set $B$ is described as a morphism from the certain combinatorial Hopf algebra of building sets to quasisymmetric functions. We define the $q$-analog $F_q(P_B)$ and derive its determining recurrence relations. The $f$-polynomial of the nestohedron $P_B$ appears as the principal specialization of the quasisymmetric function $F_q(P_B)$.
Deformation cones of graph associahedra and nestohedra
本文给出了图关联体和巢状体形变锥的刻面描述，推广了次模函数锥参数化，并构造了推广运动学关联体的运动学巢状体。
The toric g-vector of nestohedra
研究通过将简单多面体的托里克g向量表示为gamma向量的非负整数线性组合，揭示了associahedron、cyclohedron及permutahedron的托里克g向量与123-avoiding停车函数的上升统计之间的关系。
The Asymptotic Structure of Cosmological Integrals
本文基于宇宙学多面体与嵌套多面体的组合结构，分析了任意幂律FRW宇宙学中标量模型微扰积分的红外与紫外渐近行为及发散度，并利用扇区分解提取发散。
Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $Θ(m)$ on the diameter of trivially perfect graph associahedra on $m$ edge…
The method for obtaining expressions for coefficients of reverse generating functions
分析了生成函数的幂及其性质。基于整数$n$的复合的应用，引入了一类新的函数，称为复合函数。给出了求母函数倒数和逆母函数的方法，以及求解函数方程$F(A(x))=G(x)$（其中$A(x)$为未知母函数）的方法。关键词：母函数，逆，倒数，复合，方法。
Multilinear generating functions for Charlier polynomials
Charlier配置提供了Charlier多项式的组合模型。我们使用这个模型给出了一个多线性生成函数的Charlier多项式的组合证明。作为多重线性母函数的特殊情况，我们得到了Charlier多项式的双线性母函数和错位公式。
In this work we define a unified generating functions for 9 different kinds of set partitions including cyclically ordered set partitions. Such generating function depends on 4 parameters. We consider property of this function and provide combinatorial explanation for polynomials generated by this function. Two new combinatorial statistics are defined and the explicit formulae given for coefficients of parametrized polynomials defined by the generating function.
We consider the tiling generating functions of semi-hexagons and quartered hexagons with dents on their sides. In general, there are no simple product formulas for these generating functions. However, we show that the modification in the regions' width changes the tiling generating functions by only a simple multiplicative factor.
Composition of ordinary generating functions
本文对一般生成函数的合成问题提出了一种解决方法。一类新的功能，提供了一个组成的普通生成函数的介绍;主要定理提出; compositae写多项式，三角和双曲函数，指数和日志功能。结果表明，该组合适用于许多整数序列。
Redundant generating functions in lattice path enumeration
冗余母函数是一个母函数，它的项不是原问题解的一部分。我们利用冗余生成函数来研究两条路径问题。在第一个应用程序中，我们解释了一个令人惊讶的发生加泰罗尼亚数计数路径，保持低于线y = 2x。在第二个应用中，我们证明了Niederhausen和Sullivan的一个猜想。
We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.
The question of classifying the nature of the generating functions of restricted lattice walks has enjoyed much attention in past years. We prove that a certain class of octant walks have a D-finite generating function using the theory of multivariate formal Laurent series.
The Half-Perimeter Generating Function of Gated and Wicketed Ferrers diagrams
我们证明了Wicketed和Gated Ferrers图的半周长生成函数是代数的。此外，Wicketed Ferrers图的生成函数与Catalan数的生成函数密切相关。实验和证明的方法是本征传递矩阵法。
The Eigenvectors of the Right-Justified Pascal Triangle: A Shorter Proof with Generating Functions
而不是处理繁琐的二项式身份证，我们证明了卡兰的结果使用生成函数。
Proposal of a generating function of partition sequences
本文提出分拆序列的生成函数，虽其因与分拆一一对应最初看似无意义，但通过矩阵赋予系数意义并保留分拆信息，还给出适用于该生成函数的分拆限制。
The exponential generating function for the sequence A143405 in the OEIS is exp(exp(x)*(exp(x) - 1)). This paper analyzes the more general generating function exp(m*exp(b*x) + r*exp(d*x) + s) and provides asymptotics for the sequences A143405, A355291, A002872, A002874 and others in the OEIS.
In the paper, 2 explicit formulas for the Euler numbers of the second kind are obtained. Based on those formulas a exponential generating function is deduced. Using the generating function some well-known and new identities for the Euler number of the second kind are obtained.
Generating functions for plateaus in Motzkin paths
Motzkin路径中的平台是由三个步骤组成的序列：向上一步，水平一步，然后向下一步。我们发现三种不同形式的二元生成函数的高原Motzkin路径，然后推广到更长的高原。最后，我们描述了进一步的推广：一个连分数的形式，从中可以很容易地得到新的多元生成函数的各种路径统计。生成函数的几个例子给出了使用这种技术。
A triple lacunary generating function for Hermite polynomials
一些经典的正交多项式，如Hermite、Laguerre、Charlier等，已被证明是某些组合对象的生成多项式。这些组合的解释是用来证明新的身份和生成函数涉及这些多项式。在本文中，我们应用Foata的组合解释的Hermite多项式计数匹配的一组，以获得一个三重缺生成函数的Hermite多项式。我们也给出了这个生成函数的一个阴影证明。
Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences.
The $P$-partition generating function of a (naturally labeled) poset $P$ is a quasisymmetric function enumerating order-preserving maps from $P$ to $\mathbb{Z}^+$. Using the Hopf algebra of posets, we give necessary conditions for two posets to have the same generating function. In particular, we show that they must have the same number of antichains of each size, as well as the same shape (as defined by Greene). We also discuss which shapes guarantee uniqueness of the $P$-partition generating function and give a method of constructing pairs of non-isomorphic posets with the same generating fu…
Generating functions of substitutions
本文研究替换及其相关的生成函数，利用替换的递归结构分析Fibonacci替换中字母a的特征函数的根。
We present a short, direct proof of the fact that the generating function of all permutations of a fixed length $n\geq 4$ is divisible by $(1+z)^m$, where $m=\lfloor (n-2)/2 \rfloor$.
A Family of Generating Functions for Reciprocal Binomial Coefficients and Its Applications
本文提出递归二项式系数的生成函数族，推导相关恒等式并展示其在无限序列评估中的应用。
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions, including for binary and n-ary partitions.
Equality of P-partition generating functions
对于每一个标号偏序集（P，ω），我们可以为其（P，ω）-划分关联一个拟对称生成函数。我们问：什么时候两个标号偏序集有相同的生成函数？由于对应于斜舒尔函数等式的特殊情况仍然是开放的，因此（P，ω）生成函数之间的等式的完整分类可能太过期待。相反，我们确定必要条件和分离的充分条件，两个标记偏序集有相等的生成函数。我们的结论与分类的所有平等的标记偏序集与少量的线性扩展。
Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is always algebraic, and for the Kreweras step set, the generating function is always D-finite. However, apart from the particular case where the interactions are symmetric in $x$ and~$y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, we confirm their intuition that the generating function of Kreweras walks is n…
Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special cases. We give an explicit formula for this generating function in the case where $m<2k$. This result is used to obtain the generating function for the number of tilings of an $m\times n \times k$ box with $k\times k\times 1$ bricks.
Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-array trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho po…
This paper analyzes over 30 types of q-series and the asymptotic behavior of their expansions. A method is described for deriving further asymptotic formulas using convolutions of generating functions with subexponential growth. All variables in the article are integers.
In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of directed acyclic graphs and the number of strongly connected directed graphs.
We continue to consider the ordered lexicographic sequence, which is constructed according to the formal characteristics of a series of natural numbers. For analysis, we selected balanced parentheses with zeros, Motzkin words. As you know, generating functions allow you to work with combinatorial objects by analytical methods. Motzkin words are enumerated by Motzkin numbers, for the generation of which there is a corresponding generating function. In our case, restrictions are imposed on Motzkin words, for example, there are no leading zeros in bracket sets. The purpose of this article is to o…
A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of bipartite quantum systems. Direct evaluation of these generating functions is then demonstrated through the enumeration of all rooted hypermaps with up to 13 darts.
Proofs of four generating function conjectures for arbor polytopes
本文证明了四个关于poset和特定arbor序列关联聚类体不变量的生成函数猜想，包括Zeta多项式、M-三角形、Ehrhart多项式和体积函数拉普拉斯变换的闭式公式。
A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of τ
在本文中，我们介绍了一种新的方法来计算生成函数的下降和从左到右的最小值的数量在没有连续出现的模式，从1开始的排列的集合。
We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions which are evidently positive series for partitions due to Kanade and Russell. To obtain those generating functions, we first construct an evidently positive series for a key infinite product. In that construction, a series of combinatorial moves is used to decompose an arbitrary partition into a base partition together with some auxiliary partitions that bi…
We consider a generating function of the domino tilings of an Aztec rectangle with several boundary unit squares removed. Our generating function involves two statistics: the rank of the tiling and half number of vertical dominoes as in the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp. In addition, our work deduces a combinatorial explanation for an interesting connection between the number of lozenge tilings of a semihexagon and the number of domino tilings of an Aztec rectangle.
The Number of Convex Polyominoes and the Generating Function of Jacobi Polynomials
Lin和Chang给出了具有$m+1$乘$n+1$最小外接矩形的凸多项式的生成函数。Gessel表明，他们的结果意味着，这样的多项式的数量是$$ \frac{m+n+mn}{m+n}{2m+2n\choose 2m}-\frac{2mn}{m+n}{m+n\choose m}^2. $$我们表明，这一结果可以从一些二项式系数身份有关的生成函数的雅可比多项式。
Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive…
The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak composition quasisymmetric functions and Chow's quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableau…
In this paper, we consider properties of coefficients of a generating functions composition, where the outer function is a logarithmic generating function and the inner function is an ordinary generating function with integer coefficients. Using notions of composita and composition of generating functions, we get new properties for this composition. The properties can be used for distinguishing prime numbers from composite numbers. As an application, obtained results can be used to obtain new primality criteria. We obtain primality criteria for the Mersenne numbers, the Lucas numbers, the Pell…
On a property of superposition of the generating functions ln(1/(1-F(x)))
得到了生成函数ln（1/（1-F（x）的一个新的叠加性质，其中F（x）是整系数生成函数，它允许构造一个素性检验.证明了基于正数合成的定理及其推论。并给出了实例。关键词：母函数，母函数的叠加，自然数的合成.
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde matrices are derived.
Horadam sequences and their partial sums are computed via generating functions. The results are as simple as possible.
For a two parameter family of Bernoulli numbers $B_{n, p}$ the exponential generating function is derived by elementary methods.
Convolutions for Tribonacci numbers involving binomial coefficients are treated with ordinary generating functions and the diagonalization method of Hautus and Klarner. In this way, the relevant generating function can be established, which is rational. The coefficients can also be expressed. It is sketched how to extend this to Tetranacci numbers and similar quantities.
It is demonstrated how an explicit expression of the (partial) sum of Tetranacci numbers can be found and proved using generating functions and the Hadamard product. We also provide a Binet-type formula for generalized Fibonacci numbers, by explicitly factoring the denominator of their generating functions.
Generating functions for ranks of pre-modular categories
我们推导出了与量子群在单位根上相关的前模范畴的秩的生成函数。
Generating functions for moments of the quasi-nilpotent DT-operator
本文证明了DT（δ_0，1）算子T的 *-矩的某些重正化的母函数的递推公式，其中涉及形式幂级数上的运算\odot和将\odot转换为通常乘法的变换E.这个递推公式被用来证明所有这些生成函数都是有理函数，并明确地找到其中的一些。
Partitions with unique largest part and their generating functions
本研究提出最大部分唯一且其余部分构成该最大部分分拆的新型分拆，推导其生成函数并拓展至多类变体，为分拆理论研究提供新的生成函数工具。
By using an approach of the invariant theory we obtain a new formula for the ordinary generating function of the numbers of the simple graphs with $n$ nodes.
We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "$B$-ary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.
Generating functions and Abdelkader's random walk model
本文将Abdelkader提出的一类随机游走问题与Moran游走模型建立联系，揭示了两者之间的关联。
In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a relation between the rank of strongly unimodal sequences and the rank of integer partitions.
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $ζ(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such identity is then applied to show several supercongruences.
Generating function identities for $ζ(2n+2), ζ(2n+3)$ via the WZ method
本文用WZ方法给出了序列$\{ζ(2n+2)\}_{n\ge 0}, \{ζ(2n+3)\}_{n\ge 0}.$的生成函数的Koecher，Leshchiner和Bailey-Borwein-Bradley恒等式的简单证明.用同样的方法给出了这些生成函数的几种新的表示，得到了黎曼ζ函数值的快速收敛级数.
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