Recursive Formula for Binomial Determinants

• Recursive formulas for binomial determinants are defined using binomial coefficients and a Kronecker delta, establishing structured recurrence relations.
• The method employs the Desnanot–Jacobi–Dodgson identity to derive two-term recurrences that pave the way for closed-form solutions.
• These results have significant combinatorial applications, including enumerating descending plane partitions and symmetric tiling problems.

A recursive formula for binomial determinants describes the relations satisfied by a family of determinants formed from binomial coefficients, where the determinants themselves satisfy explicit recurrences. A notable instance is Andrews’s determinant, which arose in the study of descending plane partitions and the enumeration of cyclically symmetric rhombus tilings of a hexagon with holes. The evaluation of such determinants involves algebraic, combinatorial, and determinant-theoretic techniques, especially leveraging the Desnanot–Jacobi–Dodgson (DJD) identity to obtain recursive structures and closed-form expressions (Koutschan et al., 2017).

1. Definition and Family of Binomial Determinants

Let $\mu$ be an indeterminate. For integers $n \geq 1$, $s$, and $t$, define the two-parameter family of determinants: $$D_{s,t}(n;\mu) = \det_{1 \leq i,j \leq n}\bigg( \binom{\mu + i + j + s + t - 2}{j + t - 1} + \delta_{i+s,\, j+t} \bigg)$$ where $\delta_{a,b}$ denotes the Kronecker delta, which is $1$ if $a=b$ and $0$ otherwise. The determinant $D(n) \equiv D_{1,1}(n;\mu)$ is referred to as Andrews’s "curious" determinant. This family encompasses several subfamilies, each encoding enumeration problems in algebraic combinatorics, notably the enumeration of tilings of certain hexagonal regions.

2. Recurrence Relation via Desnanot–Jacobi–Dodgson Identity

A central structure is the two-term recurrence derived from the Desnanot–Jacobi–Dodgson (DJD) identity. Utilizing notational abbreviations: $n \geq 1$0 The DJD identity states: $n \geq 1$1 Dividing through by $n \geq 1$2 and isolating $n \geq 1$3 yields: $n \geq 1$4 This recursive equation encapsulates the structure of the sequence of binomial determinants and forms the backbone for their explicit evaluation [(Koutschan et al., 2017), Sec. 4, eqn. (**)].

3. Initial Data and Parity-Based Simplifications

The recursion requires specification of initial terms. Direct computation yields: $n \geq 1$5

$n \geq 1$6

A crucial structural simplification arises from parity considerations. Specifically,

$n \geq 1$7

For odd $n \geq 1$8,

$n \geq 1$9

Thus, the recursion specializes as follows:

Parity of $s$0	Structure of $s$1
Even ($s$2)	$s$3
Odd ($s$4)	$s$5

This dichotomy, demonstrated through Lemmas 3–4 in the cited work, drastically constrains the structure of the determinant sequences.

4. Derivation via Desnanot–Jacobi–Dodgson Identity

The recurrence emerges from the DJD identity as applied to infinite bi-indexed matrices of binomial coefficients with a Kronecker delta correction: $s$6 The DJD identity for such matrices can be synthesized as: $s$7 Specializing to $s$8 yields the precise two-term relation for the determinants $s$9, $t$0, $t$1, and $t$2. This recurrence, coupled with the initial conditions and even/odd behavior, uniquely determines all $t$3.

5. Closed-Form ("Single-Sum") Solution

By iterated unrolling of the recurrence and further application of DJD, a closed-form (single-sum) formula for $t$4 is provided [(Koutschan et al., 2017), Thm. 13]. Let $t$5. Then:

• For $t$6 odd,

$t$7

• For $t$8 even,

$t$9

This closed form is verified to satisfy the two-term recurrence and the requisite initial data.

6. Combinatorial Context and Significance

The determinants $$D_{s,t}(n;\mu) = \det_{1 \leq i,j \leq n}\bigg( \binom{\mu + i + j + s + t - 2}{j + t - 1} + \delta_{i+s,\, j+t} \bigg)$$0, as originally considered by Andrews, enumerate combinatorial objects—specifically, descending plane partitions and cyclically symmetric rhombus tilings of hexagons with triangular holes. The recursive evaluation and closed forms facilitate the enumeration of such tilings, resolving longstanding enumeration challenges. More broadly, the family $$D_{s,t}(n;\mu) = \det_{1 \leq i,j \leq n}\bigg( \binom{\mu + i + j + s + t - 2}{j + t - 1} + \delta_{i+s,\, j+t} \bigg)$$1 and its recursions, as analyzed by Fischer and collaborators, provide a framework for the algebraic study of tiling functions, with proofs leveraging holonomic and automated methods (Koutschan et al., 2017).

7. Broader Impact and Related Methodologies

The recursive structure provided by the DJD identity exemplifies the utility of classical determinant identities in modern combinatorics and discrete algebra. Extensions of the holonomic ansatz approach, as used in the evaluation of these binomial determinants, have wider applicability in the systematic proof and discovery of closed forms for combinatorial sums and determinants. The results connect to areas such as enumeration of plane partitions, alternating sign matrices, and symmetric tiling problems, thereby constituting a canonical example of the algebraic–combinatorial interface (Koutschan et al., 2017).