arXiv subjects

Swee Hong Chan

Publications and source records attributed to Swee Hong Chan.

At least 19 recordsLinked to original sources

arXiv2026

Equality conditions for correlation inequalities

We prove equality conditions for the Ahlswede--Daykin (AD) inequality and the Fortuin--Kasteleyn--Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Bj\"orner's and Fishburn's inequalities for linear extensions of finite posets, the Lam--Postnikov--Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede--Daykin--Schur (ADS) inequality recently introduced in Chan--Chen--Pak--Soskin (2026), which is an AD type extension of the LPP inequality.

math.CO↗

arXiv2026

Correlation inequalities for Schur positivity

We generalize the Ahlswede--Daykin inequality (1978) to a Schur positive \emph{ADS inequality}, which also contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

math.CO↗

arXiv2026

Independent Sets and Continued Fractions

Linek's 1989 problem asks whether the numbers of independent sets of trees avoid infinitely many positive integers. We show that the set of natural numbers realized as the number of independent sets of a tree has a lower growth exponent of $0.1966$. We further prove that the set of positive integers representable by connected planar graphs has asymptotic density one. Lastly, we establish a phase transition: the number of independent sets of graphs with fewer than $d|V|$ edges for any $d<1$ is contained in a set of density zero, whereas, following Shkredov's recent breakthrough on Zaremba's conjecture in continued fraction theory, there exists a constant $D$ such that the number of independent sets of graphs with at most $D|V|$ edges covers all positive integers.

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arXiv2025

Normalized matching property for competing urn models

We study the competing urn model in which $m$ balls are placed independently into $n$ urns according to (possibly distinct) ball distributions. Kahn and Neiman (2010) showed that, under identical ball distributions, the induced urn measure has \emph{conditional negative association} property and asked whether this remains true without assuming identical distributions. We answer this in the affirmative by showing that the competing urn model satisfies the \emph{normalized matching property}. This, in turn, implies conditional negative association for the induced urn measure with non-identical ball distributions, resolving the question of Kahn and Neiman.

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arXiv2025

Unimodality for Radon partitions of random vectors

Consider the (almost surely) unique Radon partition of a set of $n$ random Gaussian vectors in $\mathbb R^{n-2}$; choose one of the two parts of this partition uniformly at random, and for $0 \le k \le n$, let $p_k$ denote the probability that it has size $k$. In this paper, we prove strong unimodality results for the distribution $(p_0,\dots,p_n)$.

math.CO↗

arXiv2025

Effective resistance in planar graphs and continued fractions

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{\tau(G/e)}{\tau(G)}$, where $\tau(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedl\'a\v{c}ek's inverse problem for the number of spanning trees.

math.CO↗

arXiv2024

Spanning trees and continued fractions

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.

math.CO↗

arXiv2024

Equality cases of the Stanley--Yan log-concave matroid inequality

The \emph{Stanley--Yan} (SY) \emph{inequality} gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with $k$ fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case $k=0$. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for $k=0$, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for $k\ge 1$, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.

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arXiv2024

Linear extensions and continued fractions

We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. We apply our results to the problem of the minimum number of elements needed for a poset with a given number of linear extensions.

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arXiv2023

Linear extensions of finite posets

We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the inequalities, and the computational complexity aspects of the results.

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arXiv2023

Equality cases of the Alexandrov--Fenchel inequality are not in the polynomial hierarchy

Describing the equality conditions of the Alexandrov--Fenchel inequality has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem, and is a complexity counterpart of the recent result by Shenfeld and van Handel (arXiv:archive/201104059), which gave a geometric characterization of the equality conditions. The proof involves Stanley's order polytopes and employs poset theoretic technology.

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arXiv2023

Computational complexity of counting coincidences

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the problem, with $2\times 1 \times 1$ and $2\times 2 \times 1$ boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood--Richardson coefficients.

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arXiv2023

On the cross-product conjecture for the number of linear extensions

We prove a weak version of the cross--product conjecture: ${F}(k+1,\ell) {F}(k,\ell+1) \geq (\frac12+\varepsilon) {F}(k,\ell) {F}(k+1,\ell+1)$, where ${F}(k,\ell)$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are $k$ and $\ell$ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the {generalized cross--product conjecture}. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

math.CO↗

arXiv2023

Effective poset inequalities

We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Björner--Wachs inequality and generalize it to inequalities on order polynomials and their $q$-analogues via direct injections and FKG inequalities. Second, we give an injective proof of the Sidorenko inequality with computational complexity significance, namely that the difference is in $\#P$. Third, we generalize the Sidorenko inequality to posets with small chain intersections and give complexity theoretic applications.

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arXiv2022

Multivariate correlation inequalities for $P$-partitions

Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin--Daykin--Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell--Felsner--Trotter. The proofs are based on a multivariate generalization of the Ahlswede--Daykin inequality.

math.CO↗

arXiv2022

Correlation inequalities for linear extensions

We employ the combinatorial atlas technology to prove new correlation inequalities for the number of linear extensions of finite posets. These include the approximate independence of probabilities and expectations of values of random linear extensions, closely related to Stanley's inequality. We also give applications to the numbers of standard Young tableaux and to Euler numbers.

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arXiv2022

Recurrence of horizontal-vertical walks

Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At each discrete time step, the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability $q$). Then, it takes a mean zero horizontal step if the new label is `H', and a mean zero vertical step if the new label is `V'. This model is a randomized version of the deterministic rotor walk, for which its recurrence (i.e., visiting every vertex infinitely often with probability 1) in two dimensions is still an open problem. We answer the analogous question for the the horizontal-vertical walk, by showing that the horizontal-vertical walk is recurrent for $q \in (\frac{1}{3},1]$.

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arXiv2022

The cross-product conjecture for width two posets

The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.

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