Queue Inversion Tableaux

• Queue inversion tableaux are defined as fillings of partition diagrams with a local L-triple (quinv) statistic, crucial for expressing modified Macdonald polynomials.
• They enforce quinv-non-attacking and coquinv-sorted conditions to uniquely sort equal-height columns and cancel permutation redundancies.
• The framework connects tableau combinatorics with multiline queues and interacting particle systems, enabling precise computations of stationary probabilities.

Queue inversion tableaux are fillings of partition diagrams equipped with the queue inversion statistic $\quinv$, a local $L$-triple statistic designed to interact naturally with multiline queues, the asymmetric simple exclusion process, and the multispecies totally asymmetric zero range process. In one formulation they are arbitrary fillings $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$ used in a tableau formula for the modified Macdonald polynomial $\widetilde H_\lambda(X;q,t)$; in the compact symmetric-Macdonald setting, the relevant objects are quinv-non-attacking fillings, often further restricted to a coquinv-sorted subfamily (Mandelshtam, 6 Aug 2025, Mandelshtam, 2024).

1. Terminology and underlying fillings

The terminology is not completely uniform. One source states that it does not introduce a separate named family called “queue inversion tableaux” as a formal standalone term; instead, the precise objects are tableaux of partition shape that are quinv-non-attacking and often also coquinv-sorted, equipped with the queue inversion statistic $\quinv$ or its complement $\coquinv$ (Mandelshtam, 2024). Another source uses the term “queue inversion tableaux” directly for the tableau model attached to $\widetilde H_\lambda$ and to stationary probabilities of the multispecies totally asymmetric zero range process on a ring (Mandelshtam, 6 Aug 2025).

The ambient shape is the diagram $\dg(\lambda)$ of a partition $\lambda=(\lambda_1,\dots,\lambda_k)$, drawn as bottom-justified columns, with column $j$ of height $L$0. Rows are numbered from bottom to top and columns from left to right. A cell is $L$1, and for $L$2 one writes $L$3, while $L$4 by convention in the bottom row. A filling is a map

$L$5

and its monomial weight is

$L$6

In the compact $L$7 formula, admissibility is imposed by the quinv-non-attacking condition: for cells $L$8 and $L$9 with $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$0, the pair is quinv-attacking if either $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$1, or $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$2; a filling is quinv-non-attacking if no quinv-attacking pair carries equal entries (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).

A second restriction appears when adjacent columns have equal height. If $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$3 is $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$4-compatible, meaning $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$5, then a quinv-non-attacking filling is coquinv-sorted exactly when the top entries in those two columns increase from left to right: $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$6 This sortedness condition is the mechanism that removes the $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$7 redundancy in the compact symmetric-Macdonald formula (Mandelshtam, 2024).

2. Local geometry and the queue inversion statistic

The major index is defined by descents against the south neighbor: $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$8 where for $\sigma:\dg(\lambda)\to\mathbb Z_{>0}$9,

$\widetilde H_\lambda(X;q,t)$0

Thus $\widetilde H_\lambda(X;q,t)$1 is the number of cells above $\widetilde H_\lambda(X;q,t)$2 in its column, and bottom-row cells never contribute because $\widetilde H_\lambda(X;q,t)$3 there (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).

The queue-arm statistic uses the row below rather than the classical row of the cell itself. For $\widetilde H_\lambda(X;q,t)$4,

$\widetilde H_\lambda(X;q,t)$5

This is one of the decisive distinctions between queue inversion combinatorics and the classical Haglund–Haiman–Loehr inversion geometry (Mandelshtam, 2024).

Queue inversions are counted through $\widetilde H_\lambda(X;q,t)$6-triples. In one standard indexing, an $\widetilde H_\lambda(X;q,t)$7-triple consists of cells

$\widetilde H_\lambda(X;q,t)$8

with $\widetilde H_\lambda(X;q,t)$9. If $\quinv$0 is the top cell of its column, the triple is degenerate and the missing entry above $\quinv$1 is replaced by $\quinv$2. Writing

$\quinv$3

one defines a local predicate $\quinv$4 by declaring $\quinv$5 exactly when

$\quinv$6

An $\quinv$7-triple is then a queue inversion if $\quinv$8; otherwise it is a co-queue inversion. The global statistics are

$\quinv$9

and

$\coquinv$0

A key structural point is that the degenerate $\coquinv$1-triples record inversions among equal-height top entries (Mandelshtam, 2024).

This statistic is not a minor variant of the classical HHL inversion statistic. Queue inversions use a vertical pair together with a cell to the right in the lower row, whereas HHL inversions use the same row. Equivalently, the associated arm changes from $\coquinv$2 to $\coquinv$3. This makes $\coquinv$4 compatible with queue-theoretic constructions rather than with the original rowwise inversion geometry (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).

3. Macdonald-polynomial formulas

The basic queue-inversion tableau formula is the positive expansion

$\coquinv$5

which gives an alternative tableau formula for the modified Macdonald polynomial $\coquinv$6 (Mandelshtam, 6 Aug 2025).

When equal-height columns are sorted, this compresses to the compact formula

$\coquinv$7

where $\coquinv$8 is a $\coquinv$9-multinomial factor coming from permutations of equal-height columns. The compactness lies in replacing the full orbit of equal-height column permutations by a single sorted representative weighted by $\widetilde H_\lambda$0 (Mandelshtam, 6 Aug 2025).

For symmetric Macdonald polynomials, the compact tableau theorem takes the form

$\widetilde H_\lambda$1

This formula is the compact version of a broader non-attacking expression for $\widetilde H_\lambda$2, and it removes the prefactor $\widetilde H_\lambda$3 present in earlier unsorted formulas (Mandelshtam, 2024).

The same framework yields a formula for the integral form $\widetilde H_\lambda$4 and a non-compact formula for $\widetilde H_\lambda$5 by summing over quinv-non-attacking tableaux with product factors involving $\widetilde H_\lambda$6 and $\widetilde H_\lambda$7. A later source presents these formulas as arising from a plethystic passage from the $\widetilde H_\lambda$8 quinv formula, and also records a conjectural compact version for $\widetilde H_\lambda$9; the compact theorem itself is explicitly stated in the earlier compact-formula paper (Mandelshtam, 6 Aug 2025, Mandelshtam, 2024).

A concrete small example is given for $\dg(\lambda)$0: $\dg(\lambda)$1 For the tableau

$\dg(\lambda)$2

one has $\dg(\lambda)$3, $\dg(\lambda)$4, $\dg(\lambda)$5, and total weight

$\dg(\lambda)$6

This illustrates how $\dg(\lambda)$7, $\dg(\lambda)$8, and the $\dg(\lambda)$9-dependent product factor combine in the compact formula (Mandelshtam, 2024).

4. Multiline queues and interacting particle systems

The term “queue” comes from the relation to multiline queues and to exclusion-type particle systems. In the compact symmetric-Macdonald setting, the tableaux are in bijection with Martin’s multiline queues, and under that bijection the tableau weight becomes exactly the multiline queue weight (Mandelshtam, 2024).

For a coquinv-sorted quinv-non-attacking tableau $\lambda=(\lambda_1,\dots,\lambda_k)$0, row $\lambda=(\lambda_1,\dots,\lambda_k)$1 of the multiline queue has particles in columns

$\lambda=(\lambda_1,\dots,\lambda_k)$2

and a particle at $\lambda=(\lambda_1,\dots,\lambda_k)$3 is paired with the particle at $\lambda=(\lambda_1,\dots,\lambda_k)$4. If the pairing is nontrivial, then

$\lambda=(\lambda_1,\dots,\lambda_k)$5

and

$\lambda=(\lambda_1,\dots,\lambda_k)$6

Thus the multiline queue factor

$\lambda=(\lambda_1,\dots,\lambda_k)$7

matches the tableau factor attached to the same vertical inequality. This is the direct combinatorial explanation for the name “queue inversion”: queue inversions are the tableau-side count of the local obstructions naturally seen by queue pairings (Mandelshtam, 2024).

On the particle-system side, queue inversion tableaux compute stationary probabilities of the multispecies totally asymmetric zero range process on a ring. For $\lambda=(\lambda_1,\dots,\lambda_k)$8, the projection

$\lambda=(\lambda_1,\dots,\lambda_k)$9

is defined by

$j$0

In other words, the bottom row of the tableau records site positions, and the height of a column records the particle species. The stationary probability of a state $j$1 is

$j$2

with partition function

$j$3

This is the $j$4 specialization of the $j$5 formula (Mandelshtam, 6 Aug 2025).

A later synthesis interprets multiline queues as fermionic and the queue-inversion tableaux relevant to the zero-range process as bosonic, with the passage between them governed by plethystic substitution and compression. This suggests that the queue inversion tableau model is not merely analogous to multiline queues but is a fused tableau shadow of an infinite multiline-queue picture (Mandelshtam, 6 Aug 2025).

5. Sorting, top borders, and compactness

The compact symmetric-Macdonald formula is driven by the ordering of top entries in equal-height columns. For a quinv-non-attacking tableau $j$6, the top border is

$j$7

Within each block of equal-height columns, one defines $j$8 as the number of inversions among those top entries. A tableau is coquinv-sorted exactly when

$j$9

equivalently when $L$00 (Mandelshtam, 2024).

The decisive structural fact is that degenerate $L$01-triples contribute exactly the factor $L$02. This means that inversions among equal-height top entries are already encoded inside the queue inversion statistic itself, and compactness is achieved by selecting the unique sorted representative in each equal-height-column orbit (Mandelshtam, 2024).

The proof of compactness uses probabilistic column-swap operators $L$03 rather than the deterministic $L$04, because the latter preserve $L$05 and adjust $L$06 by $L$07 on unrestricted fillings but do not preserve non-attackingness or the $L$08-weight factors. The local detailed-balance relation

$L$09

is then iterated along a positive distinguished subexpression that sorts the top border within equal-height blocks. The total contribution of the unsorted orbit collapses to the sorted weight times

$L$10

which cancels the prefactor $L$11 in the older formula (Mandelshtam, 2024).

This sorting mechanism is the compactness phenomenon specific to queue inversion tableaux. It is not an auxiliary normalization but an intrinsic statement that the degenerate part of $L$12 measures exactly the redundancy created by permuting equal-height columns (Mandelshtam, 2024).

6. Scope, variants, and related frameworks

A common source of confusion is terminological. “Queue inversion tableaux” is not a universally fixed label for a single formally delimited class. In one paper the precise objects are described instead as sorted non-attacking tableaux endowed with the queue inversion statistic; in another, the same circle of constructions is presented directly under the name queue inversion tableaux (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).

Another common misconception is that $L$13 is simply HHL inversion in different notation. It is not. The local triples are different, the relevant arm is $L$14 rather than $L$15, the sortedness condition is imposed on equal-height top borders rather than the classical bottom-border geometry, and the resulting formulas are tailored to multiline queues, ASEP polynomials, and the multispecies totally asymmetric zero range process rather than to the original HHL setup (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).

A plausible historical antecedent is the ASEP staircase-tableau literature, where an insertion algorithm over staircase tableaux was shown to imply factorised formulas for generating polynomials and a bijection with some coloured inversion tables. This suggests an earlier inversion-table encoding in the same broad ASEP combinatorial environment, although it is a distinct tableau model (Laborde-Zubieta, 2017).

A second plausible analogue is provided by Dyck tableaux. They admit a recursive insertion algorithm, are counted by $L$16, and their history tables satisfy

$L$17

the same admissibility pattern as classical inversion-table encodings; moreover, those history tables are identified with the non-inversion tables of permutations. This suggests an insertion-based parallel rather than an identity with queue inversion tableaux (Aval et al., 2011).

In current algebraic-combinatorial usage, the subject is therefore best understood as a specific tableau realization of queue statistics on partition diagrams, sitting at the intersection of Macdonald polynomial theory, multiline queues, and interacting particle systems. Its defining content is the local $L$18-triple statistic $L$19, its compactness mechanism is coquinv-sorting of equal-height columns, and its significance lies in providing simultaneous tableau formulas for $L$20 and $L$21 together with exact stationary-probability models for exclusion and zero-range processes (Mandelshtam, 2024, Mandelshtam, 6 Aug 2025).