Papers and preprints:

Note: The papers in each subject are listed in the reverse chronological order of arXiv publication.

Order theory

  1. Multivariate correlation inequalities for P-partitions

    Swee Hong Chan and Igor Pak
    Preprint available in arXiv, 21 pp.

  2. Correlation inequalities for linear extensions

    Swee Hong Chan and Igor Pak
    Preprint available in arXiv, 23 pp.

    We employ the combinatorial atlas technology to prove new correlation inequalities for the number of linear extensions of finite posets, with applications to the numbers of standard Young tableaux and to Euler numbers.

  3. Effective poset inequalities

    Swee Hong Chan, Igor Pak, and Greta Panova
    Preprint available in arXiv, 36 pp.

    We explore inequalities on linear extensions of posets and make them effective in different ways. These inequalities include a generalization of Björner--Wachs inequality, a generalization of Sidorenko inequality, and an asymptotic version of Graham conjecture.

  4. Introduction to the combinatorial atlas

    Swee Hong Chan and Igor Pak
    Expo. Math. 40 (2022), 1014--1048.

    We introduce the method of combinatorial atlas, a new combinatorial method to prove log-concave inequalities. In particular, we give another proof of Mason conjectures for the numbers of independent sets of matroids, and the Aleksandrov--Fenchel inequality by using this method. This paper is intended to be the expository version of the paper Chan-Pak (2021).

  5. Log-concave poset inequalities

    Swee Hong Chan and Igor Pak
    Preprint available in arXiv, 71 pp.
    Extended abstract: Sem. Lothar. Combin. 86B (2022), no. 9, 12 pp. Link

    We develop a new combinatorial method to prove log-concave inequalities. Among many applications, we generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. We also rederive Stanley's inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

  6. Extensions of the Kahn--Saks inequality for posets of width two

    Swee Hong Chan, Igor Pak, and Greta Panova
    to appear in Comb. Theory, 23 pp.

    We prove a multivariate generalization of Kahn--Saks inequality for width two posets, and we characterize the condition for equality.

  7. The cross-product conjecture for width two posets

    Swee Hong Chan, Igor Pak, and Greta Panova
    Trans. Amer. Math. Soc. 375 (2022), 5923-5961.

    We provide two proofs of the cross-product conjecture of Brightwell-Felsner-Trotter (1995) for width two posets. The first proof is algebraic (matrix algebra), and the second proof is combinatorial (Lindström–Gessel–Viennot type argument).

  8. Sorting probability of Catalan posets

    Swee Hong Chan, Igor Pak, and Greta Panova
    Adv. in Appl. Math. 129 (2021), no. 102201, 13 pp.

    We show that the sorting probability of the Catalan poset has at least a polynomial decay of degree 1.25, which improves the previously known decay of degree 0.5 from my earlier work with Pak and Panova.

  9. Sorting probability for large Young diagrams

    Swee Hong Chan, Igor Pak, and Greta Panova
    Discrete Anal. (2021), no. 24, 57 pp.

    Sorting probability of a poset measures how close to 1/2 the comparison probability in 1/3-2/3 conjecture can get. We show that the sorting probability converges to 0 for posets associated to large (skew) Young diagrams with bounded number of rows, and give an asymptotic upper bound for the rate of convergence.

Abelian sandpiles and related models

  1. Abelian networks IV. Dynamics of nonhalting networks

    Swee Hong Chan and Lionel Levine
    Mem. Amer. Math. Soc. 276 (2022), no. 1358, vii+89 pp.

    We build a unifying theory for stochastic processes that exhibits the abelian property and never halts, such as sandpile process and rotor-routing process. This paper is a continuation of the abelian networks series of Bond and Levine (2016).

  2. Abelian sandpile model and Biggs-Merino polynomial for directed graphs

    Swee Hong Chan
    J. Combin. Theory Ser. A 154 (2018), 145-171.

    We show that, for a strongly-connected digraph, the generating function of recurrent configurations of the sandpile model by the number of chips is an invariant that does not depend on the choice of the sink of the sandpile model, and thus answers a conjecture of Perrot and Pham (2013).

  3. Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

    Swee Hong Chan, Henk D.L. Hollmann, and Dmitrii V. Pasechnik
    J. Algebra 421 (2015), 268-295.

    We compute the sandpile group of generalized de Bruijn graphs and generalized Kautz graphs, and in the former case we relate this group to a quotient of the group of circulant matrices over a finite field. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS.

Random walks and stochastic processes

  1. Log-concavity in planar random walks

    Swee Hong Chan, Igor Pak, and Greta Panova
    to appear in Combinatorica, 8 pp.

    We prove log-concavity of exit probabilities of lattice random walks in certain planar regions.

  2. Recurrence of horizontal-vertical walks

    Swee Hong Chan
    to appear in Ann. Inst. Henri Poincaré Probab. Stat., 34 pp.

    We study a randomized version of rotor walks on the 2D integer lattice, where the last exit from each vertex alternates between the horizontal and the vertical direction. We show that, for the uniform initial rotor configuration, this walk visits every vertex infinitely often almost surely, whereas the analogous problem for rotor walks remain an open problem.

  3. Infinite-step stationarity of rotor walk and the wired spanning forest

    Swee Hong Chan
    Proc. Amer. Math. Soc. 149 (2021), 2415 - 2428.

    We show that, for transient interger lattices, the final rotor configuration (after the rotor walker escapes to infinity) follows the law of the wired spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF, and thus answers a question raised in my previous work.

  4. A rotor configuration with maximum escape rate

    Swee Hong Chan
    Electron. Commun. Probab. 25 (2020), no. 19, 5 pp.

    We construct, for any graph, a rotor configuration for which its escape rate is equal to the escape rate of simple random walk, and thus answers a question of Florescu-Ganguly-Levine-Peres (2014).

  5. Rotor walks on transient graphs and the wired spanning forest

    Swee Hong Chan
    SIAM J. Discrete Math. 33 (2019), 2369 - 2393.

    We study rotor walks on transient graphs with initial rotor configuration sampled from the wired uniform spanning forest oriented toward infinity (OWUSF) measure. Among other things, we give a simple sufficient and necessary condition for the OWUSF measure to be a stationary distribution for the rotor walk.
    Note: The stationarity of OWUSF for transient integer lattices conjectured in Question 9.1 was later resolved positively in the follow-up to this paper.

  6. Random walks with local memory

    Swee Hong Chan, Lila Greco, Lionel Levine, and Peter Li
    J. Stat. Phys. 184 (2021), no. 6, 28 pp.

    We prove a quenched invariance principle for a class of random walks in random environment on the integer lattice, where the walker alters its own environment.

Other branches of combinatorics

  1. A bijection between necklaces and multisets with divisible subset sum

    Swee Hong Chan
    Electron. J. Combin. 26 (2019), P1.37, 18 pp.

    We construct a bijection between (1) the necklaces of length n with 2 colors, and (2) the sets of integers modulo n with subset sum divisible by n, provided that n is odd, and thus answers a bijective problem posed by Richard Stanley (Enumerative Combinatorics Vol. 1 Chapter 1, Problem 105(b)).

  2. Toric arrangements associated to graphs

    Marcelo Aguiar and Swee Hong Chan
    Sem. Lothar. Combin. 78B (2017), Art. 84, 12 pp. (Contributed talk in FPSAC 2017)

    We study toric arrangements that arise from graphs on the torus that is associated either to (1) the coroot lattice of type A or (2) coweight lattice of type A.

  3. Quasi-periodic tiling with multiplicity: a lattice enumeration approach

    Swee Hong Chan
    Discrete Comput. Geom. 54 (2015), 647-662.

    We study the multi-tiling problem by a convex polytope, where the tiling set is a finite union of translated lattices. Under a mild condition, we show that the tiling set can be replaced with a lattice, and is a step in the direction of proving the conjecture of Gravin, Robins, and Shiryaev (2012).
    Note: The conjecture mentioned above was later resolved positively by Liu in this paper.