
Stationary probabilities of the multispecies TAZRP and modified Macdonald polynomials: I (with Arvind Ayyer and James Martin).
Recently, a formula for the Macdonald polynomials P_{λ}(X;q,t) was given in terms of objects called multiline queues, which also compute probabilities of a particle model from statistical mechanics called the multispecies ASEP on a ring. It is natural to ask whether the modified Macdonald polynomials H_{λ}(X;q,t) can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative. In this paper we give a new formula for H_{λ}(X;q,t) in terms of fillings of tableaux called polyqueue tableaux. In the upcoming sequel to this paper, we show that polyqueue tableaux also compute probabilities of the multispecies totally asymmetric zero range process (mTAZRP) on a ring, and that H_{λ}(X;1,t) is equal to the partition function of the mTAZRP.

Expanding the quasisymmetric Macdonald polynomials in the fundamental basis (with Sylvie Corteel and Austin Roberts).
We derive an expansion for the quasisymmetric Macdonald polynomials in the fundamental basis of quasisymmetric functions.

Compact formulas for Macdonald polynomials and quasisymmetric Macdonald
polynomials (with Sylvie Corteel, Sarah Mason, Jim Haglund, and Lauren Williams).
We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also introduce a new quasisymmetric analogue of Macdonald polynomials. These "quasisymmetric Macdonald polynomials" refine the (symmetric) Macdonald polynomials and specialize to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and van Willigenburg.

See also FPSAC 2020 Proceedings:
Compact formulas for Macdonald polynomials and quasisymmetric Macdonald
polynomials

From multiline queues to Macdonald polynomials via the exclusion process
(with Sylvie Corteel and Lauren Williams).
Recently James Martin introduced multiline queues and used them to give a combinatorial formulas for the stationary distribution of the multispecies asymmetric simple exclusion process (ASEP) on a ring. Here we give an independent proof of Martin's result and show that by introducing additional statistics on multiline queues, we can give a new combinatorial formula for both the symmetric Macdonald polynomials P_{λ} and the nonsymmetric Macdonald polynomials E_{λ} where λ is a partition. This formula is rather different from others that have appeared in the literature. Our proof uses results of CantinideGierWheeler, who recently linked the multispecies ASEP on a ring to Macdonald polynomials.

See also FPSAC 2019 Proceedings:
From multiline queues to Macdonald polynomials via the exclusion process
This paper generalizes the results of the cylindric rhombic tableaux paper below using different methods.

Cylindric rhombic tableaux and the twospecies ASEP on a ring
(with Sylvie Corteel and Lauren Williams), to appear in
Progress in Mathematics (birthday volume for Kolya Reshetikhin).
We use some new tableaux on a cylinder called cylindric rhombic tableaux (CRT) to give a formula for the stationary distribution of the twospecies ASEP on a circle. We also use them to give a formula for Macdonald polynomials associated to partitions where all parts are 0, 1, or 2.

Toric tableaux and the inhomogeneous twospecies TASEP on a ring
, to appear in Advances in Applied Mathematics.
In this paper, I study the inhomogeneous twospecies TASEP on a ring. This is an exclusion process in which particles of different species are hopping clockwise on a ring with parameters dictating the hopping rates for different species. I introduce ``toric rhombic alternative tableaux'', which are certain fillings of tableaux on a triangular lattice tiled with rhombi, and are in bijection with the wellstudied multiline queues of Ferrari and Martin. With these tableaux I give a formula for the stationary probabilities of the twospecies inhomogeneous TASEP, which specializes to recover results of Ayyer and Linusson in the case of twospecies. I also define a Markov chain on the tableaux that projects to the twospecies TASEP, and generalizing the result from my determinantal paper below, I get an explicit determinantal formula for the probabilities.
 See also FPSAC 2018 Proceedings: Bijection from multiline queues to rhombic tableaux for the inhomogeneous 2TASEP

Combinatorics of the twospecies ASEP and Koornwinder moments
(with Sylvie Corteel and Lauren Williams),
Advances in Mathematics, 321 (2017), 160204.
Corteel and Williams introduced staircase tableaux and used them to give combinatorial formulas for steady state probabilities of the ASEP and also for AskeyWilson moments. It is wellknown that AskeyWilson polynomials can be viewed as the onevariable case of Koornwinder polynomials (also known as Macdonald polynomials of type BC). In this article we introduce rhombic staircase tableaux, and, building on previous work of Corteel and Williams, we use them to give combinatorial formulas for steady state probabilities of the twospecies ASEP and also for homogeneous Koornwinder moments. (Homogeneous Koornwinder moments are integrals of homogeneous symmetric polynomials with respect to the Koornwinder measure.) Note that rhombic staircase tableaux simultaneously generalize staircase tableaux and also the rhombic alternative tableaux from our paper with Viennot.

Rhombic alternative tableaux and assemblées of permutations
(with Xavier Viennot),
European Journal of Combinatorics, 73 (2017), 119.
The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assemblées of permutations. We describe a bijection between the rhombic alternative tableaux and these assemblées. We also provide an insertion algorithm that gives a weight generating function for the assemblées. Combined, these results give a bijective proof for the weight generating function for the rhombic alternative tableaux, which is also the partition function of the twospecies ASEP at q=1.
 See also FPSAC 2016 Proceedings: Rhombic alternative tableaux, assemblées of permutations, and the ASEP

Matrix ansatz and combinatorics of the kspecies PASEP.
In this paper I study a kspecies generalization of the twospecies PASEP in which there are k species of particles of varying weights hopping right and left on a onedimensional lattice of n sites with open boundaries. In this process, only the heaviest particle type can enter on the left of the lattice and exit from the right of the lattice. In the bulk, two adjacent particles of different weights can swap places. I prove a matrix ansatz for this model, in which different rates for the swaps are allowed. Based on this, I define ``krhombic alternative tableaux'' to give formulas for the steady state probabilities of the states of this kspecies PASEP.
This paper generalizes the results from my work with Viennot below, and also contains a Markov chain on rhombic alternative tableaux that projects to the twospecies PASEP.

Tableaux combinatorics for the twospecies PASEP
(with Xavier Viennot),
Journal of Combinatorial Theory, Series A, 159 (2017), 215239.
We study a twospecies PASEP, in which there are two types of particles, "heavy" and "light," hopping right and left on a onedimensional lattice of n cells with open boundaries. In this process, only the "heavy" particles can enter on the left of the lattice and exit from the right of the lattice. In the bulk, any transition where a heavier particle type swaps places with an adjacent lighter particle type is possible. We generalize the combinatorial results of Corteel and Williams for the ordinary PASEP by defining ``rhombic alternative tableaux'' to give a combinatorial formula for the stationary probabilities for the states of this twospecies PASEP.
This paper generalizes the q=0 result below.

MultiCatalan Tableaux and the TwoSpecies TASEP
, in
Annales d'Institut Henri Poincaré D, 3(3) (2016), 321348.
In this paper, I use the matrix ansatz to define tableaux that are certain concatenations of Catalan tableaux which I call ``multiCatalan tableaux'', to give a formula for stationary probabilities for the twospecies TASEP.
 See also FPSAC 2015 Proceedings: Tableaux combinatorics for twospecies PASEP probabilities

A Determinantal Formula for Catalan Tableaux and TASEP Probabilities
, in
Journal of Combinatorial Theory, Series A, 132 (2015), 120141.
Stationary probabilities of the TASEP with open boundaries are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with α's and β's that satisfy some conditions on the rows and columns. In this paper, I give an exact determinantal formula for the steady state probability of each state of the TASEP by constructing a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram. This result gives an α / β generalization of a formula of Narayana that counts unweighted lattice paths on a Young diagram. I also give a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with n sites, precisely k of the sites are occupied by particles. This formula is an α / β generalization of the Narayana numbers.

LGV proof of a determinantal theorem for TASEP probabilities
This is short paper with a nicer proof of the above result that directly uses the LingstromGesselViennot Lemma to get the bijection from Catalan tableaux to noncrossing lattice paths.