Binomial determinants: some closed formulae

Abstract: This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we reprove, by different means, the positivity of binomial determinants shown by Gessel and Viennot.

• The paper introduces a general recursive formula for computing binomial determinants with consecutive and nearly consecutive index sets.
• It reestablishes positivity through purely algebraic methods, linking determinants to Vandermonde structures and combinatorial interpretations.
• The study provides explicit constructions for nullspace generators and details maximal rank properties, enhancing practical computation in algebra and combinatorics.

Closed Formulae for Binomial Determinants

Introduction and Motivation

This paper systematically addresses the computation of binomial determinants indexed by sets of non-negative integers, with a focus on sets corresponding to consecutive or nearly consecutive rows or columns in the infinite binomial coefficient matrix. The results are motivated by both combinatorial and commutative algebraic questions, especially those arising from Moh's investigations into the generators of certain prime ideals in power series rings. The analysis is framed in the context of the combinatorial structure of binomial matrices, building upon and extending prior foundational results, particularly those of Gessel and Viennot.

Main Theoretical Contributions

A central result of the manuscript is a general recursive formula for binomial determinants $b_J^I = \det(B_J^I)$, where $B$ is the binomial matrix and $I$, $J$ are strictly increasing sequences of non-negative integers of equal length. Specifically, for $J \leq I$ in the coordinatewise order, the determinant decomposes as a positive rational $\pi_J^I$ times a sum of smaller binomial determinants, each indexed by subsets with reduced cardinality (Theorem~size-reduction). This recurrence forms the basis for deriving closed formulae in specialized cases.

Crucially, the paper reproves and generalizes the positivity of binomial determinants—originally established via non-intersecting lattice path bijections—by demonstrating purely algebraic arguments (Corollary~cor-positivity). This approach sidesteps the earlier combinatorial machinery and provides streamlined inductive arguments on determinant size.

For indices corresponding to consecutive columns ($J$ an interval), a closed formula connects the determinant to the Vandermonde determinant in the row indices, multiplied by a simple rational normalization involving factorials and specific binomial coefficients (Theorem~Jinterval). For consecutive rows ($I$ an interval), an alternative formula is given as a product of recursively defined rational terms $\pi_{J^{(k)}}^{I^{(k)}}$, illustrating that the determinant can always be reduced to such expressions (Theorem~Iinterval).

When both row and column indices are intervals, the determinant collapses to a single quotient of products of binomial coefficients, providing a direct algebraic interpretation of combinatorially significant quantities (Corollary~corollary-moh).

Special Configurations: Almost Consecutive Indices

The study is extended to cases where either the columns or the rows are nearly consecutive (intervals with a single element removed). For consecutive rows and almost consecutive columns, the determinants are expressed as specific binomial coefficients multiplied by an explicit rational term. Conversely, for almost consecutive rows and consecutive columns, the formula involves binomial coefficients indexed exclusively by the parameters measuring the deletion offset. These formulae provide essential tools for the explicit computation of left kernel (nullspace) generators.

Rank and Left Nullspaces of Binomial Matrices

The paper directly applies the developed determinant formulae to establish maximal-rank properties for large families of finite binomial submatrices—specifically, when the row and column indices satisfy interval or near-interval constraints (Corollary~max-rank).

A substantial section calculates the generators of the left nullspaces for binomial matrices of the form $B_J^I$ where $B$0 and $B$1 are nearly consecutive. The result is a closed-form generator, given as a signed sum of scaled binomial coefficients acting on the canonical basis, with explicit coefficients computed by the main theorems (Corollary~cor-ker). The outcomes are completely constructive, enabling direct computation in applications to algebraic or combinatorial settings.

Interchange of Rows and Columns

The authors conclude by rigorously establishing a precise algebraic relationship between the determinants $B$2 and their counterparts with row and column indices interchanged, relating the switch to explicit rational factors and providing a duality formula. This extends results of Gessel and Viennot and clarifies the symmetry inherent to binomial determinant calculations.

Implications and Potential Developments

The machinery developed in this paper has immediate implications for computational algebra—especially in describing and analyzing the structure of ideals generated by binomially indexed relations, as in Moh’s original context. The explicit closed formulae permit the direct construction of generators and nullspaces without recourse to combinatorial bijections or computational algebra systems, enhancing both theoretical understanding and algorithmic efficiency.

On a theoretical level, the unification and extension of binomial determinant formulae sharpen the interface between combinatorics and algebra, and suggest further exploration in the context of $B$3-analogues, multinomial generalizations, and applications to symmetric function theory and representation theory.

While the explicit application to AI is limited, the algebraic interpretations of binomial determinants and their nullspaces can have relevance for symbolic computation modules in computer algebra systems, error-correcting codes, and combinatorial network analysis.

Conclusion

The paper presents a comprehensive and algebraically explicit set of closed formulae for binomial determinants, generalizing combinatorial results with constructive and recursive tools. The algebraic proofs of positivity, the extension to almost consecutive index sets, and the generator construction for nullspaces collectively enrich the structural comprehension of binomial matrices and their applications in commutative algebra and combinatorics. The results are positioned as both foundational and applicable for future theoretical and computational studies involving determinants of structured combinatorial matrices.

---

Reference:

"Binomial determinants: some closed formulae" (2603.29431)