We define an algorithm on fermionic and bosonic twisted multiline queues that projects to the multispecies totally asymmetric simple exclusion process (TASEP) and the totally asymmetric zero range process (TAZRP) on a ring, respectively. Our algorithm on fermionic multiline queues generalizes the Ferrari--Martin algorithm for the TASEP, and we show it is equivalent to the algorithm of Arita--Ayyer--Mallick--Prolhac (2011). Our algorithm on bosonic multiline queues is novel and generalizes the corresponding algorithm of Kuniba--Maruyama--Okado (2016) for the TAZRP. We also define a Markov process on bosonic twisted multiline queues that projects to the TAZRP and intertwines with the symmetric group action on the rows of the multiline queues.

Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a direct sum of prime ones. In this article, we extend the notion of primeness to three generalizations of classical parking functions: vector parking functions, $(p,q)$-parking functions, and two-dimensional vector parking functions. We study their enumeration by obtaining explicit formulas for the number of prime vector parking functions when the vector is an arithmetic progression, prime $(p,q)$-parking functions, and prime two-dimensional vector parking functions when the weight matrix is an affine transformation of the coordinates.

Multiline queues are versatile objects arising from queueing theory in probability that have come to play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Together with a maj statistic, multiline queues give an elegant formula for the q-Whittacker polynomials. Generalized multiline queues (GMLQs) are obtained from the action of the symmetric group on the rows of a multiline queue. We define a generalized maj statistic on GMLQs that is preserved by this action, in order to derive a novel family of formulas, indexed by compositions, for the q-Whittaker polynomials. We define an insertion procedure on GMLQs that we call collapsing, which can be described by raising and lowering crystal operators. As an application of this procedure, one naturally recovers several classical results such as the Lascoux--Schützenberger charge formulas for q-Whittaker polynomials, Littlewood--Richardson coefficients and the dual Cauchy identity.

We give a compact tableau formula for the symmetric Macdonald polynomials P_λ in terms of a queue inversion statistic on certain sorted non-attacking tableaux. The nonsymmetric components of our formula are the ASEP polynomials, which specialize to the probabilities of the asymmetric simple exclusion process (ASEP) on a circle. Moreover, the statistics in our formula are naturally related to the dynamics of the ASEP. Our tableaux are in bijection with Martin's multiline queues (Martin 2020), from which we obtain an alternative multiline queue formula for P_λ. Finally, our formula recovers an alternative formula for Jack polynomials due to Knop and Sahi (1996).

In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type &Ctilde;. In particular, we give a combinatorial formula for the Koornwinder polynomials K_λ=K_λ(z₁,...,z_N; a,b,c,d; q,t), where λ = (1,...,1,0,...,0). We also give combinatorial formulas for all ``open boundary ASEP polynomials'' F_μ, where μ is a composition in {-1,0,1}^N; these polynomials are related to the nonsymmetric Koornwinder polynomials E_μ up to a triangular change of basis.

Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries.

We describe some recently discovered connections between one-dimensional interacting particle models and Macdonald polynomials. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomial P_λ(X;q,t) through its partition function. Through this connection, a new formula was found for P_λ by generalizing multiline queues, which were introduced by James Martin (2020) to compute stationary probabilities of the ASEP. The second particle model is the multispecies totally asymmetric zero range process (TAZRP) on a ring, which was very recently found to have an analogous connection to the modified Macdonald polynomial H_λ(X;q,t) through its partition function. This discovery coincided with a new formula for H_λ, this time in terms of tableaux with a queue inversion statistic, which also compute stationary probabilities of the TAZRP. We explain the plethystic relationship between multiline queues and queue inversion tableaux, and along the way, derive a new formula for P_λ using the queue inversion statistic. This plethystic correspondence is closely related to fusion in the setting of integrable systems.

In a previous part of this work, we gave a new tableau formula for the modified Macdonald polynomials H_λ(X;q,t), using a weight on tableaux involving the queue inversion (quinv) statistic. In this paper we explicitly describe a connection between these combinatorial objects and a class of multispecies totally asymmetric zero range processes (mTAZRP) on a ring, with site-dependent jump-rates. We construct a Markov chain on the space of tableaux of a given shape, which projects to the mTAZRP, and whose stationary distribution can be expressed in terms of quinv-weighted tableaux. We deduce that the mTAZRP has a partition function given by the modified Macdonald polynomial H_λ(X;1,t). The novelty here in comparison to previous works relating the stationary distribution of inte- grable systems to symmetric functions is that the variables x₁ ,..., x_n are explicitly present as hopping rates in the mTAZRP. We also obtain interesting symmetry properties of the mTAZRP probabilities under permutation of the jump-rates between the sites. Finally, we explore a number of interesting special cases of the mTAZRP, and give explicit formulas for particle densities and correlations of the process purely in terms of modified Macdonald polynomials.

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.

We derive an expansion for the quasisymmetric Macdonald polynomials in the fundamental basis of quasisymmetric functions.

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.

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 Cantini-deGier-Wheeler, who recently linked the multispecies ASEP on a ring to Macdonald polynomials.

This paper generalizes the results of the cylindric rhombic tableaux paper below using different methods.

We use some new tableaux on a cylinder called cylindric rhombic tableaux (CRT) to give a formula for the stationary distribution of the two-species 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.

In this paper, I study the inhomogeneous two-species 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 well-studied multiline queues of Ferrari and Martin. With these tableaux I give a formula for the stationary probabilities of the two-species inhomogeneous TASEP, which specializes to recover results of Ayyer and Linusson in the case of two-species. I also define a Markov chain on the tableaux that projects to the two-species TASEP, and generalizing the result from my determinantal paper below, I get an explicit determinantal formula for the probabilities.

Corteel and Williams introduced staircase tableaux and used them to give combinatorial formulas for steady state probabilities of the ASEP and also for Askey-Wilson moments. It is well-known that Askey-Wilson polynomials can be viewed as the one-variable 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 two-species 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.

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 two-species ASEP at q=1.

In this paper I study a k-species generalization of the two-species PASEP in which there are k species of particles of varying weights hopping right and left on a one-dimensional 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 ``k-rhombic alternative tableaux'' to give formulas for the steady state probabilities of the states of this k-species 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 two-species PASEP.

We study a two-species PASEP, in which there are two types of particles, "heavy" and "light," hopping right and left on a one-dimensional 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 two-species PASEP.

This paper generalizes the q=0 result below.

In this paper, I use the matrix ansatz to define tableaux that are certain concatenations of Catalan tableaux which I call ``multi-Catalan tableaux'', to give a formula for stationary probabilities for the two-species TASEP.

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.

This is short paper with a nicer proof of the above result that directly uses the Lingstrom-Gessel-Viennot Lemma to get the bijection from Catalan tableaux to non-crossing lattice paths.