The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated with the presence of the other edge. In this paper we give a self-contained (inductive) proof of Rayleigh monotonicity for graphs.

2008-03-31math.CO

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.

2013-03-25math.CO

Higher order log-monotonicity of combinatorial sequences

A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respectively. Finally, we not only prove that the sequence of derangement numbers is asymptotically infinitely log-monotonic, but …

2013-09-28math.CO

高斯多项式的新组合证明及 Garvan 的 $k$-秩的单调性

New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank

本文通过构造高斯多项式上的新双射，细化算法 Z，并给出广义 Rogers-Ramanujan 恒等式及 Garvan 的 $k$-秩单调性的组合证明。

2025-10-09math.CO

On monotonicity of some combinatorial sequences

We confirm Sun's conjecture that $(\root{n+1}\of{F_{n+1}}/\root{n}\of{F_n})_{n\ge 4}$ is strictly decreasing to the limit 1, where $(F_n)_{n\ge0}$ is the Fibonacci sequence. We also prove that the sequence $(\root{n+1}\of{D_{n+1}}/\root{n}\of{D_n})_{n\ge3}$ is strictly decreasing with limit $1$, where $D_n$ is the $n$-th derangement number. For $m$-th order harmonic numbers $H_n^{(m)}=\sum_{k=1}^n 1/k^m\ (n=1,2,3,\ldots)$, we show that $(\root{n+1}\of{H^{(m)}_{n+1}}/\root{n}\of{H^{(m)}_n})_{n\ge3}$ is strictly increasing.

2014-12-23math.CO

Metric graphs, cross ratios, and Rayleigh's laws

We study a notion of cross ratios on metric graphs and electrical networks. We show that several known results immediately follow from the basic properties of cross ratios. We show that the projection matrices of Kirchhoff have nice (and efficiently computable) expressions in terms of cross ratios. Finally we prove a very general version of Rayleigh's law, relating energy pairings and cross ratios before and after contracting an edge segment. As a corollary, we obtain a quantitative version of Rayleigh's monotonicity law for effective resistances. Another consequence is an explicit description…

2018-10-05math.CO

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Sums of squares and negative correlation for spanning forests of series parallel graphs

We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.

2011-10-24math.CO

Combinatorial mixed valuations

Combinatorial mixed valuations associated to translation-invariant valuations on polytopes are introduced. In contrast to the construction of mixed valuations via polarization, combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations. In particular, it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative. For combinatorially positive valuations, this has strong computational implications. Applied to the discrete volume, the results generalize and strengthen work of Bihan (2015) on discrete mixed volumes. F…

2017-08-21math.COmath.MG

关于一个在《Heady-s和Taily-s位字符串的Blimpy形状》中提出的组合猜想的证明

Proof of a combinatorial conjecture posed in "The Blimpy Shape of Heady-s and Taily-s Bit Strings"

Columbia University（哥伦比亚大学）

本文证明了关于位字符串数量的组合猜想，揭示了函数的离散凸性、负值最大参数的公式以及正期望性质。

2026-02-10math.CO

A combinatorial approach for studying LOCC transformations of multipartite states

We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangl…

2005-11-10cs.DMmath.COquant-ph

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on $\boldsymbol{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, for which the GPI is already known to ho…

2022-07-28math.COmath.PRmath.ST