Papers and preprints:

We showed that, the problem of deciding if equality occurs in Stanley-Yan log-concave matroid inequality, is not contained in the polynomial hierarchy unless polynomial hierarchy collapses to a finite level. We also completely resolved Conjecture 3.40 of Yan in Theorem 1.11.

We showed that, the problem of deciding if equality occurs in Stanley's log-concave poset inequality, is not contained in the polynomial hierarchy unless polynomial hierarchy collapses to a finite level, and thus the same conclusion applies to the more general Alexandrov-Fenchel inequality.

We explore the complexity of various coincidence problems arising naturally from combinatorics, and showed that many of these problems are not in the polynomial hierarchy unless polynomial hierarchy collapses to a finite level.

We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. An answer to Problem 7.5 and 7.6 in Kravitz - Sah is included in Corollary 1.7.

We give a broad survey of inequalities for the number of linear extensions of finite posets, with emphasis on bounds, equality conditions, and computational complexity aspects of the results.

We prove a weak version of Brightwell-Felsner-Trotter's cross--product conjecture. We also prove a version of the converse inequality and disprove the generalized cross--product conjecture.

We prove a multivariate version of Ahlswede-Daykin inequality. As applications, we prove multivariate Fishburn's correlation inequality, multivariate Daykin-Daykin-Paterson inequality and multivariate cross-product inequality for order polynomials, and Lam-Pylyavskyy inequality for Schur functions.

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.

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 Daykin--Daykin--Paterson inequality.

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).

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.

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

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).

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.

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.

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).

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).

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.

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

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.

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.

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).

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.

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.

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 Sedláček from 1969. The proof uses a connection with continued fractions and Zaremba's conjecture.

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)).

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.

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.