The Pascal matrix in the multivariate Riordan group

Abstract: We generalize the concept of Pascal matrices to matrices associated with sets of points by considering multidimensional binomial coefficients as entries. We study their properties and prove that the infinite matrix associated with the set vectors with integral coordinates is in fact an element of the multivariate Riordan group.

• The paper generalizes classical Pascal matrices to the multivariate case by constructing matrices using generalized binomial coefficients and proving their group properties via the multivariate Riordan framework.
• It develops comprehensive algebraic structures including LU decompositions, explicit inversion formulas, and exponential power representations, thereby extending classical identities to multiple variables.
• The work forges important links to Stirling polynomials and Vandermonde matrices, offering new insights for combinatorial analysis and algebraic computation in multivariate settings.

The Pascal Matrix in the Multivariate Riordan Group: An Expert Overview

Introduction and Scope

This paper presents an algebraic and combinatorial extension of classical Pascal matrices to the multivariate setting, identifying the resulting infinite matrices as elements of the multivariate Riordan group. The analysis systematically generalizes binomial coefficient structures, investigates their associated linear algebraic properties, and articulates connections to Stirling polynomials and Vandermonde matrices. The study further proves that these multivariate matrices—constructed from multidimensional binomial coefficients—admit group-theoretic characterizations analogous to those in the univariate Riordan theory, with significant implications for combinatorics and algebraic computation in several variables.

Construction of Multivariate Pascal Matrices

The foundation of this work is the definition of multivariate binomial coefficients. For $\mathbf{k}, \mathbf{i} \in \mathbb{Z}_{\geq 0}^n$, the generalized binomial coefficient is

$$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$

Matrices indexed by (possibly ordered) finite or infinite sets $R \subseteq \mathbb{Z}_{\geq 0}^n$ are formed by arranging these coefficients according to total orders compatible with the natural partial order on $\mathbb{Z}_{\geq 0}^n$ (graded reverse lexicographic preferred for block structure).

Three canonical forms are defined:

• Lower-triangular: $$L_R = (\binom{\mathbf{k}_i}{\mathbf{k}_j})_{\mathbf{k}_i,\mathbf{k}_j \in R}$$,
• Upper-triangular: $U_R = L_R^T$,
• Symmetric: $$S_R = \left(\binom{\mathbf{k}_i + \mathbf{k}_j}{\mathbf{k}_i}\right)_{\mathbf{k}_i, \mathbf{k}_j \in R}$$.

For $R = \mathbb{Z}_{\geq 0}^n$, these definitions yield infinite analogues extending the classical univariate cases.

Algebraic Structure: LU Decomposition and Determinant Formulas

A central result is that for $R$ satisfying the so-called monomial condition (i.e., $R$ coincides with standard monomials modulo some monomial ideal), several key properties of classical Pascal matrices are preserved:

• Lower and upper-triangular structure: Ordering $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$0 compatibly ensures $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$1 is lower-triangular and $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$2 is upper-triangular.
• Unit determinant: $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$3 for finite $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$4.
• Cholesky/ LU decomposition: The symmetric Pascal matrix admits a factorization $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$5.
• Matrix inversion: The inversion structure generalizes as $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$6, and $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$7 is similar to its inverse via a signed diagonal matrix $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$8.

These facts, including $$\binom{\mathbf{k}}{\mathbf{i}} = \prod_{j=1}^n \binom{k_j}{i_j}.$$9 and the explicit inversion by sign-altered binomials, extend all key classical algebraic identities to the multivariate regime under mild natural assumptions on $R \subseteq \mathbb{Z}_{\geq 0}^n$0.

Exponential Structure and Powers

The powers $R \subseteq \mathbb{Z}_{\geq 0}^n$1 (for $R \subseteq \mathbb{Z}_{\geq 0}^n$2) are shown to be exponential in the sense that

$R \subseteq \mathbb{Z}_{\geq 0}^n$3

and $R \subseteq \mathbb{Z}_{\geq 0}^n$4 can be expressed as $R \subseteq \mathbb{Z}_{\geq 0}^n$5 for a strictly lower block-bidiagonal matrix $R \subseteq \mathbb{Z}_{\geq 0}^n$6 with nonzero entries only when $R \subseteq \mathbb{Z}_{\geq 0}^n$7.

Connections to Stirling Polynomials and Vandermonde Matrices

The paper establishes a decomposition that parallels the relation between classical Pascal, Stirling, and Vandermonde matrices, but at the multivariate level. Specifically, for $R \subseteq \mathbb{Z}_{\geq 0}^n$8 satisfying the monomial condition and for polynomial sequences indexed by multidegrees, the binomial transform via $R \subseteq \mathbb{Z}_{\geq 0}^n$9 maps generalized factorial Stirling matrix sequences to Vandermonde-type matrices in several variables:

$\mathbb{Z}_{\geq 0}^n$0

where $\mathbb{Z}_{\geq 0}^n$1 (generalized Stirling polynomials of the second kind), and $\mathbb{Z}_{\geq 0}^n$2 contains powers of linear forms $\mathbb{Z}_{\geq 0}^n$3.

The Multivariate Riordan Group

The Riordan group is generalized to the multivariate context following [Cheon, Huang, Kim 2017], where elements are given by tuples $\mathbb{Z}_{\geq 0}^n$4 of invertible power series and formal variable transformations. The key result is that the infinite multivariate Pascal matrix (and all its integer powers) admit a realization as a multivariate Riordan array:

$\mathbb{Z}_{\geq 0}^n$5

This demonstrates that $\mathbb{Z}_{\geq 0}^n$6 is not only lower-triangular blockwise but captured rigorously by the group structure via the formal group law of the multivariate Riordan group.

Inverse matrices $\mathbb{Z}_{\geq 0}^n$7 and higher powers are thereby expressed analytically through compositional inverses and generating function calculus in several variables, extending classical univariate Riordan group results to arbitrary dimension.

Implications and Future Directions

The embedding of the multivariate Pascal matrix into the multivariate Riordan group has both theoretical and computational implications:

• Algebraic combinatorics: The explicit factorization and inversion formulas enable computation of combinatorial transformations and the binomial transform for multi-indexed sequences, opening avenues in multivariate generating function theory.
• Linear algebra and symbolic computation: The group structure provides tractable factorizations for block-triangular infinite matrices, with potential uses in the algebraic theory of holonomic systems, diagonalization, and explicit spectral analysis.
• Connections to representation theory and algebraic geometry: The identification of structure-preserving subsets $\mathbb{Z}_{\geq 0}^n$8 as standard monomials relates to Gröbner bases, term orders, and Hilbert function computations in commutative algebra.
• Stirling polynomials and special functions: The factorization through multivariate Stirling matrices may yield new formulas for special polynomial systems, including those critical in multivariate interpolation and asymptotic analysis.

Further research can investigate deeper structural invariants available through the multivariate Riordan group, possible connections to affine and symmetric group representations, and exploitation of these matrices in numerical schemes for multi-indexed recursions.

Conclusion

The paper rigorously generalizes Pascal matrices to the multivariate setting, proves their presence in the multivariate Riordan group, and extends key algebraic properties—including inversion, LU decomposition, and exponential powers. It demonstrates that many classical relationships (such as those involving Stirling and Vandermonde matrices) persist in the multivariate context provided suitable combinatorial compatibility conditions are imposed. This framework creates a foundation for further studies in multivariate combinatorial algebra, symbolic computation, and group representations linked to infinite lower block-triangular matrices.