Signed Multiline Queues in Interpolation Macdonald Theory

• Signed multiline queues are enhanced combinatorial objects that extend classical multiline queues by incorporating signed weights and novel pairing rules to encode interpolation Macdonald polynomials.
• They provide a detailed framework integrating algebraic recursion, queue geometry, and signed-layer mechanisms to capture the inhomogeneity and cancellations intrinsic to interpolation.
• This model bridges earlier queue constructions with modern combinatorial techniques, offering exact expansion formulas for both nonsymmetric polynomials and ASEP variants.

Signed multiline queues are a combinatorial enhancement of the classical multiline queue formalism, introduced to give a combinatorial formula for interpolation Macdonald polynomials $P_\lambda^*(x;q,t)$ and their nonsymmetric and ASEP variants. In the literal sense used in current research, a signed multiline queue is the object defined for interpolation Macdonald theory in "A combinatorial formula for Interpolation Macdonald polynomials" (Dali et al., 2 Oct 2025). Earlier multiline-queue models for multispecies TASEP, PASEP, ASEP, spectral weights, and twisted steady-state constructions use ordinary, linked, weighted, or twisted multiline queues, but not signed multiline queues in this formal sense (Ayyer et al., 2012).

1. Algebraic setting

The signed multiline queue model is attached to the interpolation Macdonald polynomial $P_\lambda^*(x;q,t)$. For a composition $\mu=(\mu_1,\dots,\mu_n)$, the relevant specialization data are

$$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$

and

$$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$

The interpolation Macdonald polynomial $P_\lambda^*(x;q,t)$ is the unique symmetric polynomial with top homogeneous part $P_\lambda$, vanishing conditions

$$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$

and normalization that the coefficient of $m_\lambda$ is $1$. The nonsymmetric interpolation Macdonald polynomial $P_\lambda^*(x;q,t)$0 is uniquely determined by

$P_\lambda^*(x;q,t)$1

Its top homogeneous component is the usual nonsymmetric Macdonald polynomial $P_\lambda^*(x;q,t)$2 (Dali et al., 2 Oct 2025).

The queue model is formulated not only for $P_\lambda^*(x;q,t)$3 but also for the interpolation ASEP polynomials. If $P_\lambda^*(x;q,t)$4 and $P_\lambda^*(x;q,t)$5 is the shortest permutation with $P_\lambda^*(x;q,t)$6, then

$P_\lambda^*(x;q,t)$7

Its top homogeneous part is the usual ASEP polynomial $P_\lambda^*(x;q,t)$8, and the symmetric interpolation polynomial decomposes as

$P_\lambda^*(x;q,t)$9

This algebraic setting explains why signs enter. Ordinary multiline queues already encode homogeneous Macdonald and ASEP polynomials. The signed refinement is designed to encode the inhomogeneous interpolation structure itself, including the extra shifts and cancellations that distinguish $\mu=(\mu_1,\dots,\mu_n)$0 from $\mu=(\mu_1,\dots,\mu_n)$1 (Dali et al., 2 Oct 2025).

2. Queue geometry and local rules

Fix a partition

$\mu=(\mu_1,\dots,\mu_n)$2

A signed multiline queue has $\mu=(\mu_1,\dots,\mu_n)$3 rows, labeled

$\mu=(\mu_1,\dots,\mu_n)$4

The unprimed rows $\mu=(\mu_1,\dots,\mu_n)$5 carry regular balls labeled by positive integers $\mu=(\mu_1,\dots,\mu_n)$6, while the primed rows $\mu=(\mu_1,\dots,\mu_n)$7 carry signed balls labeled by $\mu=(\mu_1,\dots,\mu_n)$8. The bottom row of the queue is a composition $\mu=(\mu_1,\dots,\mu_n)$9, and the queue type is $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$0 (Dali et al., 2 Oct 2025).

There are two kinds of adjacent-row pairings. In a classic layer, row $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$1 is paired with row $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$2, ignoring signs and using the same classic noncrossing and cylindrical rules as in ordinary multiline queues. In a signed layer, row $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$3 is paired with row $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$4, and the rules depend on the sign of the upper ball.

In a signed layer, pairings are made from top to bottom, in decreasing order of absolute value, and within the same absolute value from right to left. A positive ball labeled $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$5 in row $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$6 must have a ball $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$7 beneath it with $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$8; if $$k_i(\mu):=\#\{j:j<i \text{ and }\mu_j>\mu_i\}+\#\{j:j>i \text{ and }\mu_j\ge \mu_i\},$$9, it must be trivially paired straight down. A negative ball labeled $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$0 in row $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$1 has either an empty spot below it or a ball $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$2 beneath it with $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$3. Pairings in signed layers do not wrap around the cylinder; pairings in classic layers may wrap. The paper identifies this asymmetry as essential, since it is what produces the interpolation shift and the sign factors (Dali et al., 2 Oct 2025).

The local rules therefore combine two distinct mechanisms: the classical cylindrical queueing inherited from the Corteel–Mandelshtam–Williams multiline queue model, and a new signed-layer mechanism that enforces interpolation-specific admissibility constraints.

3. Weights and the role of signs

The weight of a signed multiline queue $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$4 has two factors,

$$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$5

The ball weight comes from occupied sites, and the pairing weight comes from nontrivial pairings (Dali et al., 2 Oct 2025).

For a signed row $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$6, a positive ball in column $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$7 contributes $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$8. A negative ball in row $$\widetilde{\mu}:=\left(q^{\mu_1}t^{-k_1(\mu)},\dots,q^{\mu_n}t^{-k_n(\mu)}\right).$$9 contributes

$P_\lambda^*(x;q,t)$0

Thus the sign is not an auxiliary decoration: it directly changes the monomial factor and inserts both a minus sign and a $P_\lambda^*(x;q,t)$1- and $P_\lambda^*(x;q,t)$2-shift depending on the row. For the whole queue,

$P_\lambda^*(x;q,t)$3

The pairing weights distinguish classic and signed layers. In a classic layer, a nontrivial pairing has the same weight as in the Corteel–Mandelshtam–Williams model, with statistics $P_\lambda^*(x;q,t)$4 and $P_\lambda^*(x;q,t)$5, and with a $P_\lambda^*(x;q,t)$6 factor appearing exactly in the wrap-around case. In a signed layer, the weight is simpler: $P_\lambda^*(x;q,t)$7 Here $P_\lambda^*(x;q,t)$8 counts empty positions encountered in the relevant interval. The pairing weights depend only on the absolute values of labels, not on the signs, except for the overall sign in the negative case (Dali et al., 2 Oct 2025).

The paper gives a worked example of a signed multiline queue of type $P_\lambda^*(x;q,t)$9, for which the total weight is

$P_\lambda$0

This example exhibits the characteristic mixture of monomial data, rational pairing factors, and sign contributions.

The signs are essential for three distinct reasons. They change the allowed configurations; they insert the interpolation shift through the factor $P_\lambda$1; and they produce the cancellation and inhomogeneity needed for interpolation Macdonald polynomials rather than homogeneous ones (Dali et al., 2 Oct 2025).

4. Generating functions and the main combinatorial formula

For a composition $P_\lambda$2, the signed multiline queue generating function is

$P_\lambda$3

For a partition $P_\lambda$4, the signed combinatorial partition function is

$P_\lambda$5

The main theorem is

$P_\lambda$6

and

$P_\lambda$7

Thus the interpolation ASEP polynomial is the generating function of signed multiline queues of type $P_\lambda$8, and the interpolation Macdonald polynomial is obtained by summing over all rearrangements of the parts of $P_\lambda$9 (Dali et al., 2 Oct 2025).

This is the direct signed analogue of the ordinary multiline queue formulas for $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$0 and $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$1. The relation to the older theory is structural rather than merely terminological: if all balls are regular or positive, then the signed layer pairings become trivial, and the signed multiline queue reduces to an ordinary multiline queue. In that precise sense, the signed model generalizes the earlier homogeneous queue model.

The main theorem also shows that signed multiline queues are not merely an auxiliary bookkeeping device. They provide an exact positive-and-negative weighted combinatorial expansion of the interpolation objects themselves.

5. Recursive structure, signed two-line queues, and tableau reformulation

The proof proceeds by induction on the number of row pairs. For packed compositions $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$2 of type $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$3, meaning

$$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$4

the interpolation ASEP polynomial satisfies

$$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$5

This is the interpolation analogue of the classical multiline queue recursion, but with the new factor $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$6 and the $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$7-rescaling (Dali et al., 2 Oct 2025).

To extend beyond packed compositions, the paper introduces coefficients $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$8, supported only when $$P_\lambda^*(\widetilde{\nu})=0 \qquad\text{for all partitions }\nu\neq \lambda,\ |\nu|\le |\lambda|,$$9 is a signed permutation of $m_\lambda$0, and then defines generalized signed two-line queues. If $m_\lambda$1 is such a two-line queue with bottom row $m_\lambda$2 and top row a signed permutation $m_\lambda$3 of $m_\lambda$4, its generating function is

$m_\lambda$5

The key fact is that the $m_\lambda$6 satisfy exactly the same recursion as the $m_\lambda$7. A full signed multiline queue decomposes into the bottom signed two-line piece, a classical two-line piece, and an upper signed multiline queue with one fewer row pair; this matches the algebraic decomposition of $m_\lambda$8 and completes the induction (Dali et al., 2 Oct 2025).

The same theory admits a tableau reformulation. A signed multiline queue $m_\lambda$9 is mapped to a signed queue tableau $1$0 by recording each strand as a column in a doubled diagram $1$1. The tableau model involves a major index $1$2, a coinversion count $1$3, a negative-cell count $1$4, an $1$5 statistic, and arm and leg statistics on unrestricted cells. The resulting formulas are

$1$6

The paper further defines the integral form

$1$7

and derives integrality statements for the monomial expansion coefficients of $1$8 and of $1$9. At $P_\lambda^*(x;q,t)$00, it proves the factorization

$P_\lambda^*(x;q,t)$01

using the same combinatorial machinery (Dali et al., 2 Oct 2025).

6. Relation to earlier multiline-queue models and terminological clarifications

The phrase signed multiline queues is easy to confuse with several older queue constructions. The literature represented in the cited works separates sharply into literal signed models and models that are weighted, linked, twisted, or only indirectly signed.

Paper	Queue object	Sign status
(Dali et al., 2 Oct 2025)	signed multiline queues	literal signed balls and signed pairing weights
(Ayyer et al., 2012)	Ferrari–Martin multiline queues	positive monomial weights, no $P_\lambda^*(x;q,t)$02 sign convention
(Pahuja, 2023)	linked multiline queues	$P_\lambda^*(x;q,t)$03-weighted links, no signed queue model
(Mandelshtam, 2024)	Martin’s multiline queues	signed super fillings before compression, final queue model not signed
(Aas et al., 2018)	multiline queues with spectral parameters	monomial weights and alternating Möbius inversion, no signed MLQ model
(Mandelshtam et al., 2024)	twisted multiline queues	row-permuted fermionic or bosonic queues, not signed

In "An Inhomogeneous Multispecies TASEP on a Ring" (Ayyer et al., 2012), the central objects are Ferrari–Martin multiline queues equipped with positive monomial weights and effective-rate arguments. The paper explicitly does not introduce a sign convention in the sense of $P_\lambda^*(x;q,t)$04 weights. Its closest analogue to a refined weighting is the conjectural stationary monomial

$P_\lambda^*(x;q,t)$05

together with special cases such as the three-species weight $P_\lambda^*(x;q,t)$06. These are weighted multiline queues, not signed ones.

In "Correlations in the multispecies PASEP on a ring" (Pahuja, 2023), the original Ferrari–Martin construction is replaced by linked multiline queues and a $P_\lambda^*(x;q,t)$07-bully path algorithm. The new feature is that each admissible linking choice carries a $P_\lambda^*(x;q,t)$08-dependent weight, and the total weight of all linked projections realizing a word gives its stationary probability. Again, the model is weighted rather than signed.

In "A compact formula for the symmetric Macdonald polynomials" (Mandelshtam, 2024), signs occur at the level of superized fillings, with a factor $P_\lambda^*(x;q,t)$09, but the final compact tableau formula and the alternative multiline queue formula for $P_\lambda^*(x;q,t)$10 are not signed. The cancellations happen through compression and sorting, not through a separate signed multiline queue structure.

In "Multiline queues with spectral parameters" (Aas et al., 2018), the weight of a queue is $P_\lambda^*(x;q,t)$11, and the spectral weight is a positive generating function over multiline queues. The only sign phenomenon emphasized there is the alternating-sign Boolean Möbius inversion

$P_\lambda^*(x;q,t)$12

which is not an intrinsic signed MLQ model.

In "Twisted multiline queues for the steady states of TASEP and TAZRP" (Mandelshtam et al., 2024), twisted multiline queues are composition-shaped fermionic or bosonic multiline queues, with invariance under the symmetric group action generated by row swaps $P_\lambda^*(x;q,t)$13. The twist is a row-order phenomenon, not a sign structure.

The literal notion of a signed multiline queue therefore belongs specifically to the interpolation Macdonald setting. A common misconception is to treat any weighted or refined multiline queue as “signed.” The papers above show that this is not the case: signs may appear in superization, in inclusion–exclusion, or in interpolation-specific local rules, but only (Dali et al., 2 Oct 2025) defines signed multiline queues as formal queue objects with signed balls, signed-layer constraints, and signed pairing weights.