Normal Ordering and Stirling-Type Combinatorics for Double Ore Extensions of Type (14641)

Abstract: We develop an explicit PBW normal ordering theory for the $26$ double extension regular algebras of type $(14641)$ in the Zhang-Zhang classification. With respect to the order $x_1\prec x_2\prec y_1\prec y_2$, we obtain closed two-letter formulas for the internal relations and recursive coefficient systems for mixed words, products of PBW monomials, powers of normal blocks, and noncommutative multinomial expressions. The internal coefficients are mostly quantum or skew-commutative, while the Jordan families produce Lah-Whitney, hence Stirling-type, triangular arrays. The symbolic reductions are supported by a SageMath implementation included as an ancillary file.

• The paper presents a framework for PBW normal ordering in double Ore extensions, employing closed-form formulas and recursive combinatorial reductions.
• It combines quantum and Jordan-type relations, yielding Gaussian binomials and Lah–Whitney arrays for accurate noncommutative computations.
• The work includes an algorithmic SageMath implementation that facilitates efficient computation of structure constants and noncommutative multinomials.

Normal Ordering and Stirling-Type Combinatorics for Double Ore Extensions of Type (14641)

Overview and Objectives

The manuscript "Normal Ordering and Stirling-Type Combinatorics for Double Ore Extensions of Type (14641)" (2606.08396) presents a comprehensive theory of Poincaré–Birkhoff–Witt (PBW) normal ordering for the $26$ Artin–Schelter regular double Ore extensions of Zhang–Zhang type $(14641)$. The work systematically develops explicit closed formulas and recursive combinatorics for normal ordering with respect to the order $x_1\prec x_2\prec y_1\prec y_2$. This includes two-letter internal relations, mixed generators, noncommutative multinomial expansions, and applications to computation of structure constants and block powers. The algorithmic approach is supported by a SageMath implementation, reflecting a computational and structural advance in algebraic combinatorics for noncommutative polynomial analogues.

Double Ore Extensions and the Type (14641) Classification

Double Ore extensions, as introduced by Zhang and Zhang, generalize Ore extensions to two generators added to a base Artin–Schelter regular algebra, yielding a four-generator AS-regular algebra under specific compatibility constraints. The (14641) notation refers to the graded minimal resolution of the trivial module:

$$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$

where $B$ is the double extension. The families classified in [Zhang–Zhang 2009, J. Algebra]—labeled $\mathbb{A}$ through $\mathbb{Z}$—exhibit both quantum and Jordan-type commutation, the latter being responsible for the appearance of classical triangular (Stirling-type) combinatorial arrays.

A double Ore extension $B=R_P[y_1,y_2;\sigma]$ over $R=\Bbbk_Q[x_1,x_2]$ is generated by $x_1, x_2, y_1, y_2$ with defining internal (i.e., "non-mixing") relations

$(14641)$0

and four family-specific mixed relations of the form

$(14641)$1

Normal ordering thus requires jointly handling two separate quadratic PBW systematics as well as the four explicit "crossing" rules between the $(14641)$2 and $(14641)$3 generators.

PBW Normal Ordering: Internal and Mixed Reductions

The PBW basis is indexed by monomials $(14641)$4 for $(14641)$5. The forbidden adjacent words are $(14641)$6, $(14641)$7, and all $(14641)$8 pairs. Normal ordering is realized via iterated application of the quadratic and mixed relations, ultimately expressing any product as a linear combination in PBW order.

For the internal systems, closed formulas are derived, parameterized by family (quantum or Jordan), for reductions like $(14641)$9 and $x_1\prec x_2\prec y_1\prec y_2$0. Notably, in quantum cases ($x_1\prec x_2\prec y_1\prec y_2$1), the normal ordering coefficients are Gaussian binomials ("$x_1\prec x_2\prec y_1\prec y_2$2-binomials"), whereas in Jordan-type cases ($x_1\prec x_2\prec y_1\prec y_2$3, $x_1\prec x_2\prec y_1\prec y_2$4), they result in Lah–Whitney (homogeneous Stirling) coefficients: $x_1\prec x_2\prec y_1\prec y_2$5 where $x_1\prec x_2\prec y_1\prec y_2$6 is the $x_1\prec x_2\prec y_1\prec y_2$7-rising factorial. For $x_1\prec x_2\prec y_1\prec y_2$8, $x_1\prec x_2\prec y_1\prec y_2$9 (Jordan case), this simplifies to combinations of classical binomials and rising factorials, i.e., Lah–Whitney arrays.

For mixed crossings, explicit family-wise crossing kernels $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$0 are tabulated, providing concrete reduction rules for all $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$1 families. Recursions (with initial data) are then established for:

• $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$2: the coefficients for ordering $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$3;
• $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$4: the coefficients for full mixed block orderings $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$5.

These recursions depend only on sums over the explicit family kernels and the previously discussed internal reduction arrays.

Explicit Structure Constants, Block Powers, and Noncommutative Multinomials

A significant output is the explicit computation of structure constants for PBW monomial products: $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$6 where $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$7 are determined recursively based on the kernels and internal families. These constants enable efficient calculation of all powers $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$8 by explicit recurrence.

Moreover, the framework admits noncommutative multinomial expansions for general linear forms $$0 \rightarrow B(-4) \rightarrow B(-3)^{\oplus 4} \rightarrow B(-2)^{\oplus 6} \rightarrow B(-1)^{\oplus 4} \rightarrow B \rightarrow \Bbbk \rightarrow 0,$$9, since

$B$0

with the coefficients $B$1 likewise determined by recurrences involving previously computed structure constants.

Algorithmic Implementation

All combinatorial and algebraic reductions described are algorithmically implementable. The authors supply a SageMath package ("double_ore_pbw.sage") that realizes normal ordering and all combinatorial computations for arbitrary parameters in the $B$2 families. The code uses normal forms with words as dictionary keys, and applies the described reduction system iteratively and recursively, verifying correctness of expansions, structure constants, and centrality computations.

Theoretical and Practical Implications

This work makes the machinery of PBW normal ordering in double Ore extensions manifestly explicit. From a theoretical perspective, the identification of Lah–Whitney (Stirling-type) combinatorics in Jordan-type relations elucidates a direct parallel between noncommutative algebra and classical enumerative combinatorics. The normal ordering framework here generalizes standard $B$3-analogues and reveals new combinatorial arrays tied to the deeper algebraic structure of Artin–Schelter regular "quantum" spaces.

Practically, the availability of explicit recursive combinatorics and algorithmic tools opens the way for new investigations:

• Structure constants may now be analyzed in representation-theoretic contexts.
• The explicit formulas yield arithmetic information about centers, centralizers, and discriminants for these families, especially at roots of unity.
• The recursions serve as models for extending combinatorial analysis to more complex (e.g., non-trimmed or higher-dimensional) noncommutative algebras.
• The framework facilitates computer-assisted exploration of normal forms, automorphisms, and homological phenomena.

Future Directions

Future work should include:

• Derivation of closed-form or generating function expressions for mixed reduction coefficients in sparse families.
• Examination of properties at roots of unity and implications for PI and centrality structures.
• Extension of the approach to non-trimmed double Ore extensions (with derivations and tails, leading to classical Stirling numbers).
• Development of comprehensive computational packages for PBW algebras beyond the Zhang–Zhang classification and in general skew PBW settings.

Conclusion

The paper provides a rigorous, explicit combinatorial framework for normal ordering in all double Ore extensions of type $B$4, encompassing both quantum and Jordan-type behavior. The direct connection to Stirling-type arrays and the realization of the entire machinery in explicit and computationally verifiable form constitutes a new standard for the explicit algebraic combinatorics of noncommutative PBW extensions. The results have wide-ranging applications across computational algebra, structure theory, and the combinatorics of noncommutative analogues of polynomial algebras.