Queue Inversion Statistic in Macdonald Polynomials

• Queue inversion statistic is an inversion-type measure on tableau fillings that counts specified L-shaped triples with a cyclic order rule.
• It serves as an alternative to classical inversion statistics in modified Macdonald polynomial formulas while pairing with the major index to capture key combinatorial behaviors.
• This statistic bridges combinatorial tableau theory with multispecies zero-range process models, offering insight into fusion constructions and probabilistic interpretations.

Searching arXiv for the cited paper and closely related work to ground the article in published sources. I’m unable to access the arXiv search tool in this session, so I will rely on the provided arXiv metadata and cite the supplied papers directly. The queue inversion statistic, denoted $\quinv$, is an inversion-type statistic on fillings of Young diagrams introduced to give a new tableaux formula for the modified Macdonald polynomial $\widetilde{H}_\lambda(X;q,t)$, to connect those tableaux to stationary measures of the multispecies totally asymmetric zero-range process on a ring, and, via plethystic and fusion constructions, to relate back to multiline queues and formulas for $P_\lambda(X;q,t)$ and $J_\lambda(X;q,t)$ (Mandelshtam, 6 Aug 2025). It is defined by counting specified $L$-shaped triples in a filling with a cyclic orientation rule, and it functions as a “bosonic” or multi-capacity analogue of the $\Skip$ statistic on multiline queues. Within the combinatorics of Macdonald polynomials, $\quinv$ provides an alternative to the Haglund–Haiman–Loehr inversion statistic while retaining the major index $\maj$.

1. Origin and combinatorial setting

The statistic arises in a framework that simultaneously involves Young-diagram fillings, multiline queues, and one-dimensional interacting particle systems. On the particle-system side, the relevant models are the multispecies asymmetric simple exclusion process on a ring and the multispecies totally asymmetric zero-range process on a ring. The exclusion process is the “fermionic” model, with at most one particle per position, whereas the zero-range process is its “bosonic” analogue, since a site may hold arbitrarily many particles of each species. The queue inversion statistic is attached to the latter setting (Mandelshtam, 6 Aug 2025).

For a partition $\lambda$, let $\dg(\lambda)$ denote its diagram, with columns read left-to-right and rows bottom-to-top. A filling $\widetilde{H}_\lambda(X;q,t)$0 assigns a positive integer to each cell, and its monomial weight is

$\widetilde{H}_\lambda(X;q,t)$1

The classical major index on such fillings is defined by descents: a cell $\widetilde{H}_\lambda(X;q,t)$2 not in the bottom row is a descent if $\widetilde{H}_\lambda(X;q,t)$3, and then

$\widetilde{H}_\lambda(X;q,t)$4

where $\widetilde{H}_\lambda(X;q,t)$5 is the number of cells above $\widetilde{H}_\lambda(X;q,t)$6 in the same column (Mandelshtam, 6 Aug 2025).

The queue inversion tableaux used for $\widetilde{H}_\lambda(X;q,t)$7 are thus ordinary fillings equipped with the pair of statistics $\widetilde{H}_\lambda(X;q,t)$8. In contrast with the classical multiline-queue formulas for $\widetilde{H}_\lambda(X;q,t)$9 and $P_\lambda(X;q,t)$0, the $P_\lambda(X;q,t)$1 formula using $P_\lambda(X;q,t)$2 sums over all fillings, without imposing a non-attacking constraint at that stage. The queue-oriented terminology reflects the fact that these tableaux project to states of the multispecies totally asymmetric zero-range process.

2. Definition by $P_\lambda(X;q,t)$3-triples

The basic local object counted by $P_\lambda(X;q,t)$4 is an $P_\lambda(X;q,t)$5-triple. Such a triple is either non-degenerate or degenerate. A non-degenerate $P_\lambda(X;q,t)$6-triple has cells

$P_\lambda(X;q,t)$7

so that $P_\lambda(X;q,t)$8 lies directly above $P_\lambda(X;q,t)$9, while $J_\lambda(X;q,t)$0 lies to the right of $J_\lambda(X;q,t)$1 in the same row. A degenerate triple has only the cells

$J_\lambda(X;q,t)$2

meaning that $J_\lambda(X;q,t)$3 is the top cell of its column and there is no cell $J_\lambda(X;q,t)$4 above it (Mandelshtam, 6 Aug 2025).

If the entries are $J_\lambda(X;q,t)$5, $J_\lambda(X;q,t)$6, and $J_\lambda(X;q,t)$7 in the non-degenerate case, then $J_\lambda(X;q,t)$8 is a queue inversion triple precisely when, after ordering the entries increasingly and breaking ties by reading order from top to bottom and right to left, the cyclic order is counter-clockwise. Equivalently,

$J_\lambda(X;q,t)$9

For a degenerate triple, the condition is simply $L$0. The queue inversion statistic is the total number of such triples: $L$1 This definition makes $L$2 a triple-based tableau statistic rather than a pair-based inversion count (Mandelshtam, 6 Aug 2025).

The associated non-attacking notion is adapted to this geometry. Two cells $L$3 and $L$4 with $L$5 form a quinv-attacking pair if they are in the same row, or if the right cell is one row below the left cell. A filling is quinv-non-attacking if no quinv-attacking pair has equal entries. This condition becomes important in the $L$6 and $L$7 formulas, but not in the unrestricted $L$8 expansion (Mandelshtam, 6 Aug 2025).

3. Role in formulas for Macdonald polynomials

The principal algebraic role of $L$9 is to replace the classical HHL inversion exponent in a formula for $\Skip$0. The resulting identity is

$\Skip$1

Here the sum runs over all fillings of $\Skip$2 by positive integers, $\Skip$3 is controlled by the classical major index, and $\Skip$4 is controlled by the queue inversion statistic (Mandelshtam, 6 Aug 2025).

A compact form is obtained by introducing a partial order on equal-height columns based on local queue-inversion comparisons. A filling is called $\Skip$5-sorted if, in each block of columns of equal height, the columns appear in weakly increasing order under this comparison. If $\Skip$6 is the product of the relevant $\Skip$7-multinomial coefficients for the equal-height blocks, then

$\Skip$8

The reduction from all fillings to $\Skip$9-sorted fillings is driven by local column-swapping operators that preserve $\quinv$0 and change $\quinv$1 by $\quinv$2 (Mandelshtam, 6 Aug 2025).

The same statistic also enters formulas for the integral form $\quinv$3 and the symmetric Macdonald polynomial $\quinv$4, but there it appears through its complement

$\quinv$5

With the right-arm statistic

$\quinv$6

the paper gives a formula for $\quinv$7 over quinv-non-attacking fillings and then deduces a corresponding formula for $\quinv$8, both involving $\quinv$9, $\maj$0, and factors $\maj$1 in the denominator or numerator (Mandelshtam, 6 Aug 2025).

This placement of $\maj$2 within the three Macdonald forms is structurally significant. In the $\maj$3 expansion it is the direct $\maj$4-exponent on unrestricted fillings; in the $\maj$5 and $\maj$6 expansions it is complemented, and the summation is restricted to quinv-non-attacking fillings. The paper further records a compact conjectural formula for $\maj$7 in terms of quinv-sorted fillings and notes that this conjecture has since been proved in follow-up work (Mandelshtam, 6 Aug 2025).

4. Stationary distributions of the multispecies totally asymmetric zero-range process

The probabilistic interpretation of $\maj$8 is given through the multispecies totally asymmetric zero-range process on a ring. For $\maj$9 sites and a partition $\lambda$0, a state is a multiset composition

$\lambda$1

where each $\lambda$2 is a possibly empty multiset of species and $\lambda$3. If site $\lambda$4 contains $\lambda$5 particles of species $\lambda$6 and $\lambda$7 particles of species strictly larger than $\lambda$8, then the total rate of jumps of species $\lambda$9 from site $\dg(\lambda)$0 is

$\dg(\lambda)$1

At each such jump, one particle of species $\dg(\lambda)$2 moves from $\dg(\lambda)$3 to $\dg(\lambda)$4 (Mandelshtam, 6 Aug 2025).

A filling $\dg(\lambda)$5 projects to such a state by recording, at each site $\dg(\lambda)$6, the multiset of column lengths whose bottom cell is labeled $\dg(\lambda)$7. If

$\dg(\lambda)$8

then $\dg(\lambda)$9. Under this projection, queue inversion tableaux become combinatorial representatives of zero-range configurations (Mandelshtam, 6 Aug 2025).

The stationary probability of a configuration $\widetilde{H}_\lambda(X;q,t)$00 is then

$\widetilde{H}_\lambda(X;q,t)$01

with partition function

$\widetilde{H}_\lambda(X;q,t)$02

Since the $\widetilde{H}_\lambda(X;q,t)$03 formula carries the factor $\widetilde{H}_\lambda(X;q,t)$04, setting $\widetilde{H}_\lambda(X;q,t)$05 yields

$\widetilde{H}_\lambda(X;q,t)$06

Thus the partition function of the mTAZRP of type $\widetilde{H}_\lambda(X;q,t)$07 on $\widetilde{H}_\lambda(X;q,t)$08 sites is exactly $\widetilde{H}_\lambda(X;q,t)$09, and the exponent of $\widetilde{H}_\lambda(X;q,t)$10 in the stationary weight is precisely $\widetilde{H}_\lambda(X;q,t)$11 (Mandelshtam, 6 Aug 2025).

This identification is one of the defining features of the statistic. It is not merely a tableau reformulation of a symmetric-function coefficient; it is also the explicit interaction exponent in the stationary measure of a particle system.

5. Plethystic correspondence, multiline queues, and fusion

The queue inversion statistic is introduced in a setting that also contains multiline queues for the multispecies ASEP. A multiline queue is a ball system on a cylinder together with pairings of balls between adjacent rows, subject to specific local conditions. Its weight has the form

$\widetilde{H}_\lambda(X;q,t)$12

and summing over all multiline queues of size $\widetilde{H}_\lambda(X;q,t)$13 gives $\widetilde{H}_\lambda(X;q,t)$14; multiplying by an explicit product gives $\widetilde{H}_\lambda(X;q,t)$15 (Mandelshtam, 6 Aug 2025).

The new statistic is designed as the tableaux analogue of $\widetilde{H}_\lambda(X;q,t)$16. The paper states that $\widetilde{H}_\lambda(X;q,t)$17 is conceptually “dual” to the $\widetilde{H}_\lambda(X;q,t)$18 statistic on multiline queues: $\widetilde{H}_\lambda(X;q,t)$19 counts how many balls a strand jumps over, whereas $\widetilde{H}_\lambda(X;q,t)$20 counts negative skip contributions from triples. The transformation between the multiline-queue picture and the tableau picture is mediated by the plethystic relation

$\widetilde{H}_\lambda(X;q,t)$21

together with its inverse form for $\widetilde{H}_\lambda(X;q,t)$22 (Mandelshtam, 6 Aug 2025).

Combinatorially, this plethystic passage is described as a fusion process. Infinite multiline queues with columns labeled by $\widetilde{H}_\lambda(X;q,t)$23 are grouped by forgetting the power $\widetilde{H}_\lambda(X;q,t)$24, which produces multicapacity, or bosonic, queues corresponding to queue inversion tableaux. The paper explicitly relates this to fusion in the sense of integrable systems and states that the queue inversion statistic is the combinatorial trace of this fusion (Mandelshtam, 6 Aug 2025).

This yields a clear division of roles. Multiline queues govern the fermionic, exclusion-based realization of Macdonald polynomials, while queue inversion tableaux govern the bosonic, zero-range realization. The statistic $\widetilde{H}_\lambda(X;q,t)$25 is the invariant that survives after the fermionic data encoded by $\widetilde{H}_\lambda(X;q,t)$26 has been fused into a bosonic tableau model.

6. Comparison with other inversion statistics and broader context

Within Macdonald-polynomial combinatorics, the most immediate comparison is with the Haglund–Haiman–Loehr inversion statistic. The HHL statistic uses triples of shape in which the third cell lies below the rightmost cell, whereas $\widetilde{H}_\lambda(X;q,t)$27 uses triples of shape

$\widetilde{H}_\lambda(X;q,t)$28

together with degenerate top-of-column pairs. Both statistics appear in formulas of the form

$\widetilde{H}_\lambda(X;q,t)$29

but they are not the same statistic, nor are they imposed on the same local geometry (Mandelshtam, 6 Aug 2025).

The paper also records that finding an explicit bijection on fillings preserving $\widetilde{H}_\lambda(X;q,t)$30 and sending the classical inversion statistic to $\widetilde{H}_\lambda(X;q,t)$31 was nontrivial and was posed as an open problem at the time. This emphasizes that $\widetilde{H}_\lambda(X;q,t)$32 is not merely a cosmetic reformulation of HHL inversion, but a genuinely different statistic with its own local structure and probabilistic interpretation (Mandelshtam, 6 Aug 2025).

A broader tableau-theoretic perspective comes from generalized inversion and major-index statistics on standard Young tableaux. “A generalized major index statistic on tableaux” extends the family $\widetilde{H}_\lambda(X;q,t)$33 from permutations to standard Young tableaux, with $\widetilde{H}_\lambda(X;q,t)$34 equal to the classical major index and $\widetilde{H}_\lambda(X;q,t)$35 equal to the Haglund–Stevens inversion statistic (Haglund et al., 2017). This suggests that $\widetilde{H}_\lambda(X;q,t)$36 belongs to a larger landscape of inversion-like tableau statistics, although that work does not define queue inversion triples and operates in a different combinatorial regime.

A one-dimensional antecedent for the “queue” language appears in the inversion theory of words and multiset permutations. For a multiset permutation $\widetilde{H}_\lambda(X;q,t)$37, the inversion number is

$\widetilde{H}_\lambda(X;q,t)$38

and the associated generating function is a $\widetilde{H}_\lambda(X;q,t)$39-multinomial coefficient (Mulay et al., 2018). This suggests a queue-theoretic intuition—counting out-of-order pairs in a linear arrangement—but $\widetilde{H}_\lambda(X;q,t)$40 moves beyond that setting by encoding queue-like behavior through local tableau triples rather than linear pairs. In that sense, the queue inversion statistic may be viewed as a tableau-level, bosonic refinement of inversion counting adapted to Macdonald polynomials and the mTAZRP.

The worked examples in the source paper illustrate these features concretely. For the filling of shape $\widetilde{H}_\lambda(X;q,t)$41 shown there, the weight is

$\widetilde{H}_\lambda(X;q,t)$42

so $\widetilde{H}_\lambda(X;q,t)$43 and $\widetilde{H}_\lambda(X;q,t)$44 (Mandelshtam, 6 Aug 2025). In a smaller probabilistic example with $\widetilde{H}_\lambda(X;q,t)$45 and $\widetilde{H}_\lambda(X;q,t)$46, the preimage of a specified mTAZRP configuration under the projection map consists of six tableaux, and the sum of their weights gives the stationary numerator for that configuration. These examples are representative of the statistic’s dual role: it is simultaneously a tableau statistic, a Macdonald-polynomial exponent, and a stationary-weight exponent for a zero-range process (Mandelshtam, 6 Aug 2025).