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arxiv: 2607.01141 · v1 · pith:WD4VSM6Ynew · submitted 2026-07-01 · 🧮 math.CO

Normal ordering in the (p,q)-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

Toufik Mansour , Lahcen Oussi , Matthias Schork This is my paper

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keywords (p,q)-deformed Weyl algebranormal orderingrook placementsStirling numbersstaircase boardsdeformed rook numberscombinatorial interpretations
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Normal ordering in the (p,q)-deformed generalized Weyl algebra corresponds to weighted (p,q)-deformed s-rook placements on staircase boards.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the normal ordering process defined by the (p,q)-commutation relations XY - q YX = h Y^s Z_p, XZ_p = p Z_p X, and Z_p Y = p Y Z_p in the algebra generated by X, Y, Z_p produces coefficients that define a new model of (p,q)-deformed s-rook numbers. These numbers supply explicit combinatorial interpretations of the associated (p,q)-generalized Stirling numbers as rook placements on staircase boards. A sympathetic reader would care because the construction extends classical and recent rook-theoretic interpretations of Stirling numbers to the general case p ≠ 1 while preserving the algebraic structure for arbitrary s. The work thereby links noncommutative algebra directly to enumerative combinatorics on boards.

Core claim

In the (p,q)-deformed generalized Weyl algebra generated by X, Y, and Z_p satisfying the given commutation relations with s a nonnegative integer, the normal ordering process defines (p,q)-deformed s-rook numbers. These numbers furnish combinatorial interpretations of the (p,q)-generalized Stirling numbers via rook placements on staircase boards, extending several classical and recent formulations to the general p ≠ 1 setting.

What carries the argument

(p,q)-deformed s-rook numbers derived from the normal ordering process in the algebra.

Load-bearing premise

The coefficients produced by applying the commutation relations to normal-order monomials can be consistently identified with the weights of rook placements on staircase boards for arbitrary p, q, and s.

What would settle it

A direct computation of the coefficient of a normal-ordered term such as X Y^k Z_p for small fixed s, p, q that fails to equal the corresponding weighted rook count on the associated staircase board.

Figures

Figures reproduced from arXiv: 2607.01141 by Lahcen Oussi, Matthias Schork, Toufik Mansour.

Figure 1
Figure 1. Figure 1: The Ferrers board B33211 with a 4-file placement which is not a 4-rook placement. Example 3.1. If the board Bλ has ℓ columns, then clearly fkpBλq “ rkpBλq “ 0 if k ą ℓ. Furthermore, f1pBλq “ r1pBλq “ |λ|, the number of boxes. For k “ ℓ, we have fℓpBλq “ śℓ j“1 λj since we can choose the boxes in the ℓ columns independently. Let us consider the staircase board Jn “ Bpn´1q¨¨¨321. It is well known (see, e.g.,… view at source ↗
Figure 2
Figure 2. Figure 2: The Ferrers board B33211 with a 3-rook placement with 1-row creation rule. Following Goldman and Haglund [14], we define the i-rook numbers as follows. Definition 3.2. Let i P N0. Given a Ferrers board B, the k-th i-rook number r piq k pBq is the number of ways to place k non-attacking rooks on the board B going from right to left, creating i new rows to the left of each rook. Clearly, for i “ 0 one recove… view at source ↗
Figure 3
Figure 3. Figure 3: Augmented Ferrers board associated to XZ2XZ3Y ZXY XXZ4Y . A word ω in the letters X, Y and Z is in normal ordered form if the order of the letters is Y lZ mXn, see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Augmented Ferrers board associated to the normal ordered word Y lZ mXn. letter Z in ω corresponds to moving the labelled circle on the path outlined by ω. In the example of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The augmented Ferrers board from [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Normal ordering the word Y XY XY X using augmented Ferrers boards. p111q Ø q 3 p21q Ø pqh p12q Ø pqh p112q Ø q 2h p22q Ø h 2 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Rook placements on the Ferrers board of Y XY XY X. ‚ A box lying to the left of the rook and in the column of another rook does not contribute a factor of p since this column is cancelled by the other rook to the left. We now formalize the above observations and introduce a pp, qq-weight for rook placements which generalizes the one for p “ 1 from the preceding section (for s “ 0). Let ϕ be a placement of … view at source ↗
Figure 8
Figure 8. Figure 8: A Ferrers board with a 6-rook placement and marking of the boxes ac￾cording to their class. Now, we can define the pp, qq-deformed rook numbers as follows. Definition 4.3. Let B be a Ferrers board and let RpB, kq be the set of placements of k non-attacking rooks on B. Then the pp, qq-deformed rook numbers are defined by (23) Rh;p,qrB, ks :“ ÿ ϕPRpB,kq ωp,qpB; ϕq “ h k ÿ ϕPRpB,kq p #p´boxesq #q´boxes . Note… view at source ↗
Figure 9
Figure 9. Figure 9: Normal ordering Y XY XY X using augmented Ferrers boards ps “ 2q. We now introduce, for s ą 0, certain pp, qq-weights for s-rook placements which generalize the one for p “ 1 introduced by Celeste et al. [8] and considered in the preceding section, see (18). Let us denote again a rook placement satisfying the s-row creation rule by ϕ, and the set of all rook placements of k rooks satisfying the s-row creat… view at source ↗
Figure 10
Figure 10. Figure 10: The Ferrers board B8875533311 with a 5-rook placement for s “ 2 (left) and the equivalent representation with boxes marked by their class (right). In analogy to (18) (for s ą 0 but p “ 1) and (23) (for s “ 0 but arbitrary p) we define the associated pp, qq-deformed s-rook numbers as follows. Definition 4.11. Let s ą 0. Let B be a Ferrers board and let RspB, kq be the set of placements of k rooks on B sati… view at source ↗
Figure 11
Figure 11. Figure 11: pp, qq-weights of rook placements on the Ferrers board of pY Xq 3 (s “ 2). Proposition 4.18. Let wm,n,u be an arbitrary word in the letters X, Y and Zp satisfying the commu￾tation relations (8) of the pp, qq-deformed generalized Weyl algebra As;h|p,q. Then one has the normal ordering result (42) wm,n,u “ |mÿ |´m1 k“0 p |u|p|m|´kq`L pm,n,uqRs,h;p,qrBw ˚m,n,u , ks Y |n|`ps´1qkZ |u|`k p X|m|´k , where w ˚ m,… view at source ↗
read the original abstract

In this paper, we investigate the combinatorial structure arising from the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$, and $Z_p$, satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, where $s\in \mathbb{N}_0$. Our primary objective is to use the normal ordering process defined by these relations to develop a novel model of $(p, q)$-deformed rook theory. Specifically, we introduce a new framework of $(p, q)$-deformed $s$-rook numbers derived from this normal ordering process. Utilizing these combinatorial models, we provide explicit combinatorial interpretations for the associated $(p, q)$-generalized Stirling numbers via rook placements on staircase boards. Our results extend several classical and recent formulations in the literature to the general $p\neq 1$ setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that the normal-ordering coefficients in the (p,q)-deformed generalized Weyl algebra generated by X, Y, Z_p under the relations XY - q YX = h Y^s Z_p, XZ_p = p Z_p X, and Z_p Y = p Y Z_p arise as weighted (p,q)-deformed s-rook numbers on staircase boards; these in turn supply explicit combinatorial interpretations for the associated (p,q)-generalized Stirling numbers, extending prior work to the case p ≠ 1.

Significance. If the algebraic-to-combinatorial identification holds for arbitrary p, q, s, the work supplies a new parameter-dependent rook-theoretic model for deformed Stirling numbers, generalizing classical and recent rook interpretations in a uniform way. The derivation of the deformed s-rook numbers directly from the normal-ordering process is a notable strength when rigorously verified.

minor comments (2)
  1. [Abstract] Abstract: the claim of 'explicit combinatorial interpretations' is central; the main text should display the explicit mapping (e.g., the precise weighting rule for the (p,q)-deformed s-rook numbers) with at least one fully worked small example for concrete p, q, s values.
  2. Because the manuscript is labeled 'II', a brief recap of the algebra definition and normal-ordering procedure from part I would improve readability without lengthening the paper substantially.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; new combinatorial model constructed from algebra

full rationale

The derivation begins with explicitly stated (p,q)-commutation relations for the algebra generated by X, Y, Z_p. Normal ordering coefficients are then used to define (p,q)-deformed s-rook numbers on staircase boards, which in turn interpret the associated generalized Stirling numbers. This is a direct construction rather than a reduction to fitted inputs or prior self-citations. The abstract and description contain no load-bearing self-citation chains, no renaming of known results as new derivations, and no self-definitional loops where the target quantities are presupposed in the inputs. The reader's assessment of score 2 aligns with a minor (non-load-bearing) self-citation possibility from part I, but the central claim remains independently derived from the given relations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the three stated commutation relations, the existence of a well-defined normal-ordering procedure, and the assumption that the resulting coefficients admit a rook-placement interpretation on staircase boards. No additional free parameters beyond p, q, s, h are introduced in the abstract.

free parameters (1)
  • p, q, s, h
    Deformation parameters and scaling factor appearing in the commutation relations that define the algebra.
axioms (1)
  • domain assumption The generators satisfy XY - q YX = h Y^s Z_p, XZ_p = p Z_p X, and Z_p Y = p Y Z_p.
    These relations are taken as the definition of the (p,q)-deformed generalized Weyl algebra.
invented entities (1)
  • (p,q)-deformed s-rook numbers no independent evidence
    purpose: To count the coefficients arising in normal ordering via weighted rook placements.
    New combinatorial objects introduced to model the algebra.

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