Fan Ge, Gongxiang Liu
In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $γ_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite dihedral group.
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Let $Γ$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n \geq 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D} := \{(\begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix}) \mid a \in \mathbb{Z}\}$. In [5] it was asked whether $Γ$ admits a nowhere-identity $D_{2n}$-flow if and only if it admits a nowhere-identity $\mathbb{D}$-flow with $|a| < n$ (a `nowhere-identity $\mathbb{D}_n$-flow'). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of the existence of nowhere-identity $\ma…
The Harborth constant of a finite group $G$, denoted $\gs(G)$, is the smallest integer $k$ such that the following holds: For $A\subseteq G$ with $|A|=k$, there exists $B\subseteq A$ with $|B|=\exp(G)$ such that the elements of $B$ can be rearranged into a sequence whose product equals $1_G$, the identity element of $G$. The Harborth constant is a well studied combinatorial invariant in the case of abelian groups. In this paper, we consider a generalization $\gs(G)$ of this combinatorial invariant for nonabelian groups and prove that if $G$ is a dihedral group of order $2n$ with $n\ge 3$, then…
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of distributions.
Distance-regular Cayley graphs over (pseudo-) semi-dihedral groups
本文对半二面体群和伪半二面体群上的距离正则 Cayley 图给出部分刻画，这两类群均为含指数 2 循环子群的 2-群，回应了 Miklavič 和 Potočnik 提出的刻画问题。
Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the int…
In a recent paper, Caracciolo, Sokal and Sportiello presented, inter alia, an algebraic/combinatorial proof for Cayley's identity. The purpose of the present paper is to give a "purely combinatorial" proof for this identity; i.e., a proof involving only combinatorial arguments together with a generalization of Laplace's Theorem, for which a "purely combinatorial" proof is already known.
We give a combinatorial proof of Guo's multi-generalization of Munarini's identity, answering a question of Guo.
We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.
Given a subset $S$ of the non-identity elements of the dihedral group of order $2m$, is it possible to order the elements of $S$ so that the partial products are distinct? This is equivalent to the sequenceability of the group when $|S| = 2m-1$ and so it is known that the answer is yes in this case if and only if $m>4$. We show that the answer is yes when $|S| \leq 9$ and $m$ is an odd prime other than 3, when $|S| = 2m-2$ and $m$ is even or prime, and when $|S| = 2m-2$ for many instances of the problem when $m$ is odd and composite. We also consider the problem in the more general setting of …
In this note we present a combinatorial proof of an identity involving poly-Bernoulli numbers and Genocchi numbers. We introduce the combinatorial objects, $m-$barred Callan sequences and show that the identity holds in a more general manner.
In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes and multinomial identities are given.
We explain how the identity $$\sum_{i+j=n}\binom{2i}{i}\binom{2j}{j}\;=\;\displaystyle4^n$$ is an easy consequence of the inclusion-exclusion principle.
We examine an identity originally stated in Ramanujan's ``lost notebook'' and first proven algebraically by Andrews and combinatorially by Kim. We give two independent combinatorial proofs and interpretations of this identity, which also extends an identity recently proven by Pak and Waarnar related to the product of partial theta functions: First, we give a direct combinatorial proof, using the involution principle, of a special case of the identity, and extend this into a direct combinatorial proof of the full identity as written. Second, we show that the identity can be rewritten, using min…
Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over $G$ (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithme…
Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g \in G \}$. A finite unordered sequence of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. We denote by $\mathsf s (G)$ (or $\mathsf E (G)$ respectively) the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$ respectively). In this paper, we provide the exact values of $\mathsf s (G)$ and $\mathsf E (G)$ for Dihedral and Dicyclic groups a…
Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_ω(n)$ and $p_ν(n)$ introduced by Andrews, Dixit and Yee. In this paper, we present a combinatorial proof of an interesting identity in their work.
The Alladi-Gordon identity plays an important role for the Alladi-Gordon generalization of Schur's partition theorem. By using Joichi-Stanton's insertion algorithm, we present an overpartition interpretation for the Alladi-Gordon key identity. Based on this interpretation, we further obtain a combinatorial proof of the Alladi-Gordon key identity by establishing an involution on the underlying set of overpartitions.
In his paper, "On a Partition Function of Richard Stanley," George Andrews proves a certain partition identity analytically and asks for a combinatorial proof. This paper provides the requested combinatorial proof.
The title identity appeared as Problem 75-4, proposed by P. Barrucand, in Siam Review in 1975. The published solution equated constant terms in a suitable polynomial identity. Here we give a combinatorial interpretation in terms of card deals.
Let $f(m,c)=\sum_{k=0}^{\infty} (km+1)^{k-1} c^k e^{-c(km+1)/m} / (m^kk!)$. For any positive integer $m$ and positive real $c$, the identity $f(m,c)=f(1,c)^{1/m}$ arises in the random graph theory. In this paper, we present two elementary proofs of this identity: a pure combinatorial proof and a power-serial proof. We also proved that this identity holds for any positive reals $m$ and $c$.
The $n$-th rencontres number with the parameter $r$ is the number of permutations having exactly $r$ fixed points. In particular, a derangement is a permutation without any fixed point. We presents a short combinatorial proof for a weighted sum derangement identities.
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
In this paper we give a combinatorial proof and refinement of a Rogers-Ramanujan type partition identity of Siladić arising from the study of Lie algebras. Our proof uses generating functions and $q$-difference equations.
A combinatorial proof of an identity involving Eulerian numbers
本文通过研究膨胀超单纯形的alcoved三角剖分，给出了涉及欧拉数的恒等式的一个组合证明，并描述了标准单纯形三角剖分的对偶图，推测了一般情形下的结构，同时组合地证明了欧拉数等于超单纯形的归一化体积。
Recently N.Jing discovered a certain combinatorial identity from validity of the Serre relations in some vertex representations of quantum Kac-Moody algebras. We generalize this identity, in particular, extending it from polynomials to elliptic functions, and interprete the obtained identities in terms of tensor products of evaluation representations of the quantum loop algebra $U_q(\tilde{gl_2})$ or the elliptic quantum group $E_{ρ,γ}(sl_2)$.
Combinatorial interpretation of a truncated identity of Gauss
本文针对高斯恒等式的截断版本，利用过划分中最小排除非划线部分的概念，给出了新的组合解释，丰富了截断theta级数的组合研究。
A binomial coefficient identity due to Zhi-Wei Sun is the subject of half a dozen recent papers that prove it by various analytic techniques and establish a generalization. Here we give a simple proof that uses weight-reversing involutions on suitable configurations involving dominos and colorings. With somewhat more work, the method extends to the generalization also.
Hodge-Newton indecomposability and a combinatorial identity
本文通过统一证明展示Hodge-Newton不可分解性在解释一个源自有限Coxeter部分的仿射Deligne-Lusztig变种的组合恒等式中的价值。
Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicycli…
A Combinatorial Proof of an Identity of Malik and Sarma on Overpartitions with Repeated Smallest Non-Overlined Part
本文为Malik和Sarma提出的关于具有重复最小非上划线部分的超分拆的四个恒等式中剩余的一个提供了组合证明，解决了该开放问题。
Combinatorial proof of a permuted basement Macdonald polynomial identity
研究置换基 MacDonald 多项式恒等式，通过将 Concha - Lapointe 恒等式细化到特定子族，给出组合证明，并揭示其与 Kazhdan - Lusztig 对合的等价关系。
Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a generalized Worpitzky identity $$\binom{an+b}{r}^p = \sum_{i=0}^{rp} A_{a,b,r}(p,i) \binom{n+rp-i}{rp},$$ and asked whether there is a combinatorial interpretation of the numbers $A_{a,b,r}(p,i)$. We provide such a combinatorial interpretation by defining a notion of descents in colored multipermutations, and then proving that $A_{a,b,r}(p,i)$ is equal to th…
Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the combinatorial objects corresponding to a sum of positive terms, we establish bijections that lead a telescoping relation. We illustrate this idea by giving a combinatorial telescoping relation for a classical identity of MacMahon. Recently, Andrews posed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials which was derived bas…
Recently, Defant and Propp [2020] defined the degree of noninvertibility of a function $f\colon X\to Y$ between two finite nonempty sets by $\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. We obtain an exact formula for the expected degree of noninvertibility of the composition of $t$ functions for every $t\in \mathbb{N}$. An equivalent formulation for the definition of the degree of noninvertibility is then the starting point for a generalization yielding a seemingly new combinatorial identity involving the Stirling transform of the signed Stirling numbers of the first kind.
A combinatorial proof of Jacobi's elliptic identity via alternating permutations
本文通过交替排列方法，结合Entringer数和Dumont-Viennot蛇，为雅可比椭圆恒等式提供了一个新的组合证明。
We derive a combinatorial identity which is useful in studying the distribution of Fourier coefficients of L-functions by allowing us to pass from knowledge of moments of the coefficients to the distribution of the coefficients.
A unicellular map, or one-face map, is a graph embedded in an orientable surface such that its complement is a topological disk. In this paper, we give a new viewpoint to the structure of these objects, by describing a decomposition of any unicellular map into a unicellular map of smaller genus. This gives a new combinatorial identity for the number $ε_g(n)$ of unicellular maps of size $n$ and genus $g$. Contrarily to the Harer-Zagier recurrence formula, this identity is recursive in only one parameter (the genus). Iterating the construction gives an explicit bijection between unicellular maps…
For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for $r\in \mathbb{N}\cup\{0\},s\in \mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $\mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $\underlineλ$, which is a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to that for …
We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function methodology of modern enumerative combinatorics and their applications to computational problems.
We give an elementary proof of an interesting combinatorial identity which is of particular interest in graph theory and its applications. Two applications to enumeration of forests with closed-form expressions are given.
Consider the elements of the group algebra CS_{n} given by R_{j}=Sigma_{i=1}^{j-1}(ij), for 2<=j<=n. Jucys [3 - 5] and Murphy[7] showed that these elements act diagonally on elements of S_{n} and gave explicit formulas for the diagonal entries. As requested by the late S. Kerov, we give a combinatorial proof of this work in case j=n and present several similar results which arise from these combinatorial methods.
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it esta…
Characterizing Transfer Systems for Non-Abelian Groups
本文针对非阿贝尔群的转移系统展开研究，明确计算三类非阿贝尔群的宽度，扩充了涵盖多类典型非阿贝尔群的转移系统格集合，为同伦组合研究提供实用资源。
Some notes on an identity of Frisch
本文利用 Frisch 的组合恒等式证明并推广了涉及调和数的若干经典恒等式，并进一步推导出关于奇调和数的新组合恒等式。
A Short Combinatorial Proof of the Pons-Batle Identity for Counting Tree-Child Networks
针对计数树-子网络的Pons-Batle恒等式，本文给出一个简短的组合证明，明确了相关网络数目满足的递推关系。
Consider a pyramid made out of unit cubes arranged in square horizontal layers, with a ledge of one cube's length around the perimeter of each layer. For any natural number $k$, we can count the number of ways of choosing $k$ unit cubes from the pyramid such that no two cubes are in the same horizontal layer; we can also count the number of ways of choosing $k$ unit cubes from the pyramid such that no two cubes come from the same vertical slice (taken parallel to a fixed edge of the pyramid) {\it or} from two adjacent slices. Djakov and Mityagin first established, using functional analysis, th…
Kronecker coefficients via the Giambelli identity for Schur functions
本文通过结合Littlewood的Schur函数恒等式和Giambelli恒等式，提出了一种研究Kronecker系数的框架，将问题转化为涉及钩形索引的交替和，并应用于特定分区的Kronecker系数的组合解释。
In this paper we prove some combinatorial identities which can be considered as generalizations and variations of remarkable Chu-Vandermonde identity. These identities are proved by using an elementary combinatorial-probabilistic approach to the expressions for the $k$-th moments ($k=1,2,3$) of some particular cases of recently investigated discrete random variables. Using one of these Chu-Vandermonde-type identities, two combinatorial congruences are established.
Combinatorial and algebraic proofs of Keller's $A_2\square A_2$ $q$-dilogarithm identity
该研究针对与平方积箭图$A_2\square A_2$相关的凯勒$q$-双对数恒等式，通过生成函数、带色分拆双射、四色弧代数三种方法，以迭代Durfee分解为核心完成了组合与代数证明，补充了该恒等式的新证明途径。
In this paper we give combinatorial proofs of some well known identities and obtain some generalizations. We give a visual proof of a result of Chapman and Costas-Santos regarding the determinant of sum of matrices. Also we find a new identity expressing permanent of sum of matrices. Besides, we give a graphical interpretation of Newton-Girard identity.
In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix. Finally, we use these results to derive various combinatorial identities.
New Polynomial Identities and Some Consequences
本文借助 Euler Beta 函数与二项式定理推导两个多项式恒等式，推广已知结果，并由此得到及推广 Frisch、Klamkin 组合恒等式，还给出一类恒等式的推导方案。
In this paper, we give a new bijective proof of a multiset analogue of even-odd permutations identity. This multiset version is equivalent to the original coin arrangements lemma which is a key combinatorial lemma in the Sherman's Proof of a conjecture of Feynman about an identity on paths in planar graphs related to combinatorial solution of two dimensional Ising model in statistical physics.
Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard-Riordan--like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between…
Combinatorial approach to Andrews-Gordon and Bressoud type identities
本研究受Warnaar工作启发，通过构造整数分拆相关集合间显式保权重保长度双射，给出Andrews-Gordon与Bressoud恒等式和式侧的组合解释，还发现并证明了奇偶条件相反的Kurşungöz恒等式的q级数版本，同时证明了一批该类已知及新恒等式。
We obtain a finite form of Jacobi's identity and present a combinatorial proof based on the structure of synchronized partitions.
Combinatorial relations among restricted and half Eulerian polynomials of types $A$, $B$, and $D$
本文研究了类型 A、B 和 D 的欧拉多项式之间的组合关系，推导了多项恒等式并解决了开放性问题。
The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [arXiv:1503.05339] and for this generalization the stationary measure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.
In their recent book on combinatorial identities, Quaintance and Gould devoted one chapter to Melzak's identity. We give new proofs for this identity and its generalization.
Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups
研究二面体群上4度Cayley图的自同构群与结构，通过分析不同生成集的Cayley图性质，确定其分类与构造方法。
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