Tony C. Dorlas, Alexei L. Rebenko, Baptiste Savoie
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.
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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.
The classical Dodgson identity can be interpreted as a quadratic identity of spanning forest polynomials, where the spanning forests used in each polynomial are defined by how three marked vertices are divided among the component trees. We prove an analogous result with four marked vertices.
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.
This paper discusses the enumeration for the total number of all rooted spanning forests of the labeled complete tripartite graph. We enumerate the total number by a combinatorial decomposition.
Lajos Takacs gave a somewhat formidable alternating sum formula for the number of forests of unrooted trees on $n$ labeled vertices. Here we use a weight-reversing involution on suitable tree configurations to give a combinatorial derivation of Takacs' formula.
The Dodgson/Muir Identity is an identity on determinants of matrix minors that generalizes the Dodgson Identity. Using the matrix tree theorem, we create an equivalent forest identity on ordered sets of $k$ forests. We then prove the generalized Forest Identity, and by extension the Dodgson/Muir Identity, using an edge-swapping involution. This algorithm is a generalization of the Red Hot Potato algorithm, developed by the author in 2021 to prove the Dodgson Identity.
The Lewis Carroll Identity expresses the determinant of a matrix in terms of subdeterminants obtained by deleting one row and column or a pair of rows and columns. Using the matrix tree theorem, we can convert this into an equivalent identity involving sums over pairs of forests. Unlike the Lewis Carroll Identity, the Forest Identity involves no minus signs. In 2011, Vlasev and Yeats suggested that such a Forest Identity could be proven using edge transfers similar to Zeilberger's 1997 matrix proof. However, until now, such an algorithm has not yet been developed. In this paper, we provide thi…
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.
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.
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.
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…
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.
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.
Combinatorial Courant-Fischer-Weyl Minimax Principle on Cheeger $k$-constants of Weighted Forests
本文为加权森林的 Cheeger $k$-常数建立组合极小极大原理，证明森林 1-Laplacian 特征值独立性，提出改进高阶 Cheeger 不等式，并给出 $h_k=λ_k(Δ_1)$ 的组合证明。
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.
The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous--Broder Theorem
该研究提出有根森林的序质量恒等式，以此直接证明无需可逆性等假设的扩展Aldous--Broder定理，并在Lean4中形式化相关内容，为图上随机生成森林的采样提供支撑。
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.
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.
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 $f(m,c)=\sum_{k=0}^{\infty} (km+1)^{k-1} c^k e^{-c(km+1)/m} / (m^kk!)$. 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 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.
Bounded Littlewood identity related to alternating sign matrices
本文建立了与交错符号矩阵相关的Littlewood恒等式推广的有界版本，并给出其两边的组合解释，旨在最终构造组合证明以理解交错符号梯形与平面分割对象的关系。
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.
Irreducible characters in the symmetric group are of special interest in combinatorics. They can be expressed either combinatorially with ribbon tableaux, or algebraically with contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called "trace forest", with which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that gi…
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…
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。
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.
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions, and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
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。
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…
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.
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 …
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.
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.
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。
Forest formulas of discrete Green's functions
本文通过枚举树和森林给出离散格林函数的组合解释，建立有向图和加权图的森林公式，并据此推导命中时间等图不变量的组合证明。
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…
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。
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.
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…
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。
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。
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。
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a sin…
This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.
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。
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.
For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.
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.
We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of …
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.
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。
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。
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