Pandimensional Recurrences

• Pandimensional recurrences are multi-indexed extensions of classical recurrence relations that underpin multidimensional combinatorial structures and space-filling mappings.
• They enable unique Zeckendorf decompositions and hyper-Catalan enumerations while employing block-digit expansion methods for constructing space-filling curves.
• These recurrence frameworks support limit laws, summand minimality, and bijective mappings between ℝ and ℝ^d, with broad applications in combinatorics and geometric analysis.

A pandimensional recurrence is a recurrence relation formulated to operate in an unbounded number of dimensions or indexed by multi-indices, generalizing classical univariate recurrences. These frameworks provide a unified approach to multidimensional combinatorial structures, unique base decompositions in higher-rank modules, and recursive generation of multivariate mappings such as space-filling curves. Such recurrences are foundational for multidimensional Zeckendorf representations, hyper-Catalan structures, Geode coefficients, and bijective mappings between $\mathbb{R}$ and $\mathbb{R}^d$ via space-filling functions.

1. Formal Definitions and Core Examples

Pandimensional recurrences extend the notion of recurrence relations from the univariate to the multivariate or "multi-indexed" context, often indexed by infinite tuples or lattice points. Notable frameworks include:

• Multidimensional Linear Recurrences: Given an integer $k\geq 2$ and a coefficient vector $c=(c_1,\ldots,c_k)$ satisfying $c_1 \geq \cdots \geq c_k=1$, define for $n\in\mathbb{Z}$ a sequence of $X_n\in\mathbb{Z}^{k-1}$ by
  • $X_0=0$
  • $X_{-i}=e_i$, $1\leq i\leq k-1$, where $\mathbb{R}^d$0 are standard basis vectors,
  • $\mathbb{R}^d$1 for $\mathbb{R}^d$2.
  • This is the basis for general multidimensional Zeckendorf-type decompositions (Cheng et al., 8 Oct 2025).
• Hyper-Catalan Recurrences: Hyper-Catalan numbers $\mathbb{R}^d$3 are defined over arbitrary multi-indices $\mathbb{R}^d$4 and satisfy a convolution-style recurrence extracted from a multivariate generating function equated as a zero of a polynomial:

$\mathbb{R}^d$5

where $\mathbb{R}^d$6 is a multivariate formal power series (Rubine, 6 Jul 2025).

• Space-Filling Function Recurrences: The mapping from $\mathbb{R}^d$7 to $\mathbb{R}^d$8 (Peano, Hilbert, and generalizations) is constructed as a sequence of recurrences parameterized by the dimension $\mathbb{R}^d$9 and side length $k\geq 2$0, utilizing serpentine Hamiltonian paths and block-digit expansion (Jaffer, 2014).

2. Multidimensional Zeckendorf and Linear Recurrence Decompositions

The core principle is that every vector $k\geq 2$1 in $k\geq 2$2 has a unique representation as a sum of non-adjacent $k\geq 2$3-recurrence vectors. For coefficient vectors $k\geq 2$4 satisfying $k\geq 2$5, the decomposition is governed by the following:

• c-Representation: A finite sequence $k\geq 2$6 of nonnegative integers is a $k\geq 2$7-representation for $k\geq 2$8 if $k\geq 2$9.
• c-Satisfying Representation (c-SR): A $c=(c_1,\ldots,c_k)$0-representation that satisfies constraints generalizing "no two consecutive Fibonacci indices"—formally, non-adjacency generalized for the recurrence structure.

Existence and uniqueness of such representations are established by normalizing any representation using "carrying" and "borrowing" moves, which modify the coefficients without changing the sum, decreasing a monotone associated with the total number of summands (Cheng et al., 8 Oct 2025). These moves terminate finitely under the weakly decreasing $c=(c_1,\ldots,c_k)$1 condition, providing a unique canonical form.

This construction generalizes 1D Zeckendorf's theorem and allows transfer of PLRS properties—such as summand minimality and Gaussian behavior of summand counts—to the multidimensional setting via explicit bijections.

3. Hyper-Catalan, Geode Recurrences, and Multivariate Convolutions

Pandimensional recurrences for hyper-Catalan numbers $c=(c_1,\ldots,c_k)$2 and Geode coefficients $c=(c_1,\ldots,c_k)$3 arise from the structure of their multivariate generating function:

• The convolution-style recurrence for the hyper-Catalans, extracted by equating the generating function as a polynomial root, is given as:

$c=(c_1,\ldots,c_k)$4

where the indices are over all arrangements of $c=(c_1,\ldots,c_k)$5 partitioned into $c=(c_1,\ldots,c_k)$6 parts (Rubine, 6 Jul 2025).

• The Geode coefficients satisfy a "lesser-sum" recurrence, expressing each $c=(c_1,\ldots,c_k)$7 as an integer combination of hyper-Catalans of related and strictly smaller indices, and recursively in terms of itself and known $c=(c_1,\ldots,c_k)$8-terms.

Special closed forms are available for selected multi-index families, including Fuss-Catalan and bi-tri cases.

Properties of these recurrences:

• Multi-index convolution structure generalizes classical Catalan recurrences to arbitrary dimensions.
• The generating-series root identity encompasses all component relations in a single formal equation.
• A closed-form and combinatorial interpretation for general Geode coefficients remains open.

4. Pandimensional Space-Filling Function Recurrences

Pandimensional recurrences play a central role in the construction of bijective continuous mappings from the unit interval to the $c=(c_1,\ldots,c_k)$9-cube (space-filling curves and functions). The framework (Jaffer, 2014) proceeds via:

• Integer Recurrences: The forward map $c_1 \geq \cdots \geq c_k=1$0 recursively constructs multidimensional grid points via block-digit expansion in base $c_1 \geq \cdots \geq c_k=1$1, employing alignment and permutation operations dependent on serpentine Hamiltonian paths.
• Inverse Recurrences: The map $c_1 \geq \cdots \geq c_k=1$2 inverts the above via re-alignment and Hamiltonian lookup, ensuring mutual invertibility at the discrete level.
• Continuous Limits: The continuous maps $c_1 \geq \cdots \geq c_k=1$3 and $c_1 \geq \cdots \geq c_k=1$4 are realized as limits of these integer recurrences or as non-terminating geometric series, providing absolutely convergent mappings and their inverses.
• Scaling and Centering: The self-similar structure enables extension beyond the unit cube and recentering on arbitrary lattice points.

This formalism uniformly generates Peano, Hilbert, and isotropic space-filling curves across arbitrary rank $c_1 \geq \cdots \geq c_k=1$5 and cell size $c_1 \geq \cdots \geq c_k=1$6, supporting analytic evaluation of isotropy and dimensionality reduction performance.

Concept	Recurrence Structure	Indexing Domain
Multidim. Zeckendorf	Linear, weakly dec. $c_1 \geq \cdots \geq c_k=1$7	$c_1 \geq \cdots \geq c_k=1$8
Hyper-Catalan	Multivariate convolution	$c_1 \geq \cdots \geq c_k=1$9
Geode	Hyper-Catalan + recursive sub	$n\in\mathbb{Z}$0
Space-Filling Functions	Block-digit, path-aligned	$n\in\mathbb{Z}$1

5. Properties and Derived Theorems

Pandimensional recurrences inherit and generalize limit laws and extremal properties from their univariate ancestors:

• Limit Laws: The distribution of the number of summands in multidimensional Zeckendorf representations, when sampled uniformly from a suitably defined region, converges to a Gaussian as region size increases (Cheng et al., 8 Oct 2025).
• Summand Minimality: Under the sharp condition $n\in\mathbb{Z}$2, the normalized multidimensional representations are summand-minimal (Cheng et al., 8 Oct 2025).
• Space-Filling Performance: Hilbert-type curves, constructed via these recurrences, are asymptotically isotropic and yield lower dimension-reduction distortion than Peano or non-precessing variants, e.g., Hilbert exponent $n\in\mathbb{Z}$3 vs. Peano $n\in\mathbb{Z}$4 in 2D (Jaffer, 2014).
• Convolution and Factorization: The multivariate generating-function root/factorization, e.g., $n\in\mathbb{Z}$5, encapsulates both hyper-Catalan and Geode recurrences (Rubine, 6 Jul 2025).

Classical theorems such as Lekkerkerker’s, gap distributions, and large deviations for scalar PLRS directly lift to the multidimensional context via bijection arguments.

6. Special Cases and Generalizations

• Fuss-Catalan and Bi-Tri Geode Closed Forms: Explicit hypergeometric products for select index vectors provide direct calculations in special cases, e.g., single-shape (Fuss) and bi-tri cases for Geode coefficients (Rubine, 6 Jul 2025).
• f-decompositions: Generalizations where the forbidden gap before allowable summands is position-dependent, extending beyond conventional c-recurrences (Cheng et al., 8 Oct 2025).
• Centered, Scaled Space-Filling Curves: Extension of finite-granularity recurrences to all of $n\in\mathbb{Z}$6 via centering and periodic continuation (Jaffer, 2014).

7. Context, Limitations, and Open Problems

• Sharpness of Conditions: The weakly-decreasing requirement on $n\in\mathbb{Z}$7 for Zeckendorf uniqueness is necessary; otherwise, normalization procedures can fail to terminate (Cheng et al., 8 Oct 2025).
• Lack of General Geode Closed Form: While recurrences allow expansion of Geode coefficients in terms of hyper-Catalans, a general closed form and combinatorial interpretation for arbitrary $n\in\mathbb{Z}$8 remain open (Rubine, 6 Jul 2025).
• Continuity and Inverse Pathologies: The inverse space-filling functions are not continuous on a measure-zero set of points where digit expansions are not unique (Jaffer, 2014).
• Unified Generating Functions: The use of multivariate generating-function roots as encoding devices provides a compact mechanism for deriving an entire system of interrelated recurrences.

Pandimensional recurrences serve as organizing principles for multidimensional combinatorial enumeration, canonical vector decompositions, and recursive generation of mappings, enabling unified analysis and algorithmic computation across growing families of multidimensional mathematical structures.