Geode Numbers in Combinatorics

• Geode numbers are multi-indexed combinatorial numbers derived from hyper‐Catalan generating series, serving as coefficients that encode explicit solutions to algebraic equations.
• They exhibit diverse computational behavior, with closed forms available in bivariate cases and holonomic recurrences enabling efficient computation in higher dimensions.
• Geode numbers carry varied combinatorial interpretations—from lattice path enumerations to refined ordered tree statistics—enhancing their significance in enumerative combinatorics.

Geode numbers are multi-indexed combinatorial numbers derived from hyper-Catalan numbers and from a factorization of the hyper-Catalan generating series. Introduced by Norman Wildberger and Dean Rubine in connection with explicit solutions to algebraic equations, they arise as coefficients of a quotient polynomial obtained after dividing a hyper-Catalan generating polynomial by the linear form $t_1+\cdots+t_k$, and, in the infinite-variable formulation, as coefficients of the Geode series $\mathbf{G}$ defined by $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$ with $\mathbf{S}_1=t_1+t_2+\cdots$ (Amdeberhan et al., 14 Aug 2025, Amdeberhan et al., 22 Jun 2025). Their definition is elementary, but their behavior is highly nonuniform across dimensions: the bivariate case admits closed forms, the three-variable case is accessible through holonomic recurrences, and the four-variable diagonal case became an explicit computational challenge that was later resolved by a specialized recursive computation (Amdeberhan et al., 14 Aug 2025, Rubine, 25 Dec 2025).

1. Formal definition and generating-series framework

For a list of nonnegative integers $[m_1,\ldots,m_k]$, the hyper-Catalan number is defined by

$$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$

If $n=m_1+\cdots+m_k$, the associated generating polynomial is

$$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$

Wildberger and Rubine proved that $P_{n,k}(t_1,\ldots,t_k)$ is divisible by $t_1+\cdots+t_k$, and the geode polynomial is then

$\mathbf{G}$0

The Geode number $\mathbf{G}$1 is the coefficient of $\mathbf{G}$2 in $\mathbf{G}$3 (Amdeberhan et al., 14 Aug 2025).

A closely related formulation uses the formal power series $\mathbf{G}$4 satisfying

$\mathbf{G}$5

together with $\mathbf{G}$6. In this setting Wildberger and Rubine found the factorization

$\mathbf{G}$7

where $\mathbf{G}$8 is the Geode series and its coefficients $\mathbf{G}$9 are the Geode numbers (Amdeberhan et al., 22 Jun 2025). In the supplied literature these two presentations coexist: one emphasizes quotients of finite generating polynomials, the other emphasizes a factorized infinite-variable series.

Concrete coefficients can already be very large. A recorded example is

$\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$0

computed using the authors’ Maple package (Amdeberhan et al., 14 Aug 2025).

2. Hyper-Catalan background and explicit low-dimensional formulas

The hyper-Catalan numbers provide the ambient structure from which Geode numbers are extracted. In the roofed-polygon notation, the hyper-Catalan number

$\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$1

counts subdivisions of a roofed polygon into $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$2 triangles, $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$3 quadrilaterals, $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$4 pentagons, and so forth (Rubine, 6 Jul 2025). The same coefficients also enumerate ordered trees of prescribed degree sequence in the tree-theoretic formulation (Gossow, 24 Jul 2025).

The bivariate Geode numbers admit an explicit closed form. For $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$5,

$\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$6

This formula was one of the conjectures posed by Wildberger and Rubine and later proved by Amdeberhan and Zeilberger; it also appears in related work of Rubine and in the later computational survey (Amdeberhan et al., 22 Jun 2025, Amdeberhan et al., 14 Aug 2025). Because of this closed form, large instances such as $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$7 can be computed in milliseconds (Amdeberhan et al., 14 Aug 2025).

Several special families beyond the basic bivariate case are also explicit. If only one polygon size occurs, then the Geode number reduces to a Fuss-type value: $\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$8 If two consecutive shapes occur, one has

$\mathbf{S}-1=\mathbf{S}_1\mathbf{G}$9

For two distinct, not necessarily consecutive, shapes,

$\mathbf{S}_1=t_1+t_2+\cdots$0

These formulas establish that a substantial part of the low-dimensional and sparse-support theory can be reduced to factorial products or alternating hyper-Catalan sums, even though no general closed form is known (Rubine, 6 Jul 2025).

3. Combinatorial interpretations

One major development in the subject is the appearance of independent combinatorial models for the same coefficients. Gessel gave a lattice-path interpretation based on the formal power series $\mathbf{S}_1=t_1+t_2+\cdots$1, the Geode $\mathbf{S}_1=t_1+t_2+\cdots$2, and the auxiliary series $\mathbf{S}_1=t_1+t_2+\cdots$3, proving

$\mathbf{S}_1=t_1+t_2+\cdots$4

In this model, $\mathbf{S}_1=t_1+t_2+\cdots$5 is the excursion generating function, $\mathbf{S}_1=t_1+t_2+\cdots$6 counts nonnegative paths, and $\mathbf{S}_1=t_1+t_2+\cdots$7 counts positive paths. The coefficients of $\mathbf{S}_1=t_1+t_2+\cdots$8, hence the Geode numbers, enumerate nonnegative lattice paths with prescribed step weights (Gessel, 12 Jul 2025).

A distinct ordered-tree interpretation was later provided by Gossow. Let $\mathbf{S}_1=t_1+t_2+\cdots$9 be an ordered rooted planar tree with specified numbers of nodes of each out-degree. Then the coefficient of $[m_1,\ldots,m_k]$0 in $[m_1,\ldots,m_k]$1 equals the number of ordered pairs $[m_1,\ldots,m_k]$2 such that $[m_1,\ldots,m_k]$3 is a leaf of $[m_1,\ldots,m_k]$4 and $[m_1,\ldots,m_k]$5 is visited before any non-leaf node in post-order traversal (Gossow, 24 Jul 2025). This interpretation is more refined than simply counting trees of a given type.

That refinement matters because an earlier conjecture, attributed to Wildberger and Rubine, was disproved. The incorrect claim was that Geode numbers count ordered trees with prescribed internal node degrees and one extra leaf. Gossow’s result shows that the correct statistic is not “all extra leaves,” but only those leaves lying in the initial segment of the post-order traversal before any internal node appears (Gossow, 24 Jul 2025). This episode is central to the subject’s early history: it illustrates that Geode numbers are not merely a transparent augmentation of hyper-Catalan objects, but encode a subtler positional statistic.

4. Holonomic structure and three-dimensional computation

The transition from two to three variables marks a sharp increase in complexity. The available sources state that no simple closed form exists for $[m_1,\ldots,m_k]$6, but that these coefficients satisfy second-order pure linear recurrences with polynomial coefficients in each variable. Schematically,

$[m_1,\ldots,m_k]$7

with analogous recurrences in the other coordinate directions (Amdeberhan et al., 14 Aug 2025).

The diagonal sequence $[m_1,\ldots,m_k]$8 is especially structured: it satisfies a second-order linear recurrence in $[m_1,\ldots,m_k]$9,

$$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$0

where $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$1 and $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$2 are rational functions of degree $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$3 in $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$4. These recurrences make values such as $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$5 rapidly computable (Amdeberhan et al., 14 Aug 2025).

The methodology used to obtain such recurrences is explicitly described as a combination of experimental mathematics and the holonomic ansatz. Initial terms are generated directly from the definition; guessing algorithms based on linear algebra and undetermined coefficients are then used to find recurrences; once conjectured, the recurrences can be rigorously proved using the holonomic systems approach. The supplied sources emphasize that these functions are holonomic by definition, and that standard computer algebra, including Koutschan’s Mathematica package, can formally verify the resulting recurrences (Amdeberhan et al., 14 Aug 2025). This suggests that Geode numbers occupy a region of enumerative combinatorics where exact symbolic structure exists, but must often be discovered computationally before it can be proved.

5. Four-dimensional difficulty and the computation of $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$6

The first computational survey stated the four-dimensional case much more starkly. Its central claim was that the methods used in three dimensions do not scale to four dimensions: no recurrence for $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$7 had been found, direct computation from the definition was described as utterly infeasible for large arguments, and a donation of $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$8 US dollars to the OEIS was offered for the first computation of the four-dimensional diagonal term $$C(m_1,\ldots,m_k)=\frac{(2m_1+3m_2+\cdots +(k+1)m_k)!\,(1+m_1+2m_2+\cdots +km_k)!}{m_1!\cdots m_k!}.$$9 (Amdeberhan et al., 14 Aug 2025).

Later work reported that this value had in fact been computed, and claimed the reward. In that paper the Geode polynomial $n=m_1+\cdots+m_k$0 is defined by

$n=m_1+\cdots+m_k$1

with diagonal elements $n=m_1+\cdots+m_k$2. Coefficient extraction from the factorization yields the Geode recurrence

$n=m_1+\cdots+m_k$3

and hence

$n=m_1+\cdots+m_k$4

with base case $n=m_1+\cdots+m_k$5 (Rubine, 25 Dec 2025).

The reported computation did not rely on a new diagonal holonomic recurrence. Instead, it used memoization, slice-by-slice evaluation, and ratio formulas for neighboring hyper-Catalan values, such as

$n=m_1+\cdots+m_k$6

where $n=m_1+\cdots+m_k$7 and $n=m_1+\cdots+m_k$8 are the total edge and vertex counts defined in the paper (Rubine, 25 Dec 2025). The paper states that each slice involves $n=m_1+\cdots+m_k$9 Geode numbers, there are $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$0 slices, hence $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$1 Geode elements are computed, and that after accounting for large-integer arithmetic the total time is $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$2. The specific computation of $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$3 reportedly took approximately 36 hours on a standard PC, and the resulting integer has 6303 digits (Rubine, 25 Dec 2025).

6. Later generalizations and open directions

The Geode also has a noncommutative extension. In the framework of noncommutative symmetric functions, if $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$4 is the Lagrange series solving

$$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$5

then for $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$6 the series

$$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$7

has coefficients that are polynomials in the $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$8 with non-negative integer coefficients, and these are again called geode numbers (Novelli et al., 23 Nov 2025). Under the specialization $$P_{n,k}(t_1,\ldots,t_k)=\sum_{m_1+\cdots+m_k=n} C(m_1,\ldots,m_k)t_1^{m_1}\cdots t_k^{m_k}.$$9, the coefficients refine ordered trees, \L ukasiewicz words, Dyck words, nondecreasing parking functions, and noncrossing partitions. The same paper gives Gessel’s formula in the noncommutative setting,

$P_{n,k}(t_1,\ldots,t_k)$0

and develops $P_{n,k}(t_1,\ldots,t_k)$1-geodes and $P_{n,k}(t_1,\ldots,t_k)$2-Lagrange or Schröder-type analogues (Novelli et al., 23 Nov 2025).

Open problems remain prominent in the commutative theory. The survey literature states that explicit closed forms for higher dimensions remain unknown, that recurrences for higher-dimensional diagonals were for some time out of reach because of exponential combinatorial complexity, and that the general structure and properties of Geode numbers in higher dimensions form a fertile domain for further research (Amdeberhan et al., 14 Aug 2025). A further historical nuance is that one 2025 paper still said that, except for special cases, what general Geode coefficients count combinatorially was unknown (Rubine, 6 Jul 2025), whereas another paper from the same year supplied an ordered-tree interpretation correcting a previous conjecture (Gossow, 24 Jul 2025). This chronological progression indicates that the subject developed unusually rapidly: algebraic definitions and computational recurrences appeared first, while satisfactory combinatorial interpretations emerged only afterwards and only after a mistaken first guess had been eliminated.

Geode numbers therefore sit at the intersection of algebraic equations, Lagrange inversion, polygon dissections, ordered trees, lattice paths, and holonomic computation. Their simplest instances resemble Catalan and Fuss-Catalan numbers, but the higher-dimensional theory is computationally harder and structurally less transparent. That combination of elementary definition, rapid growth of complexity, and multiple competing models is precisely what has made the Geode an active topic in contemporary enumerative combinatorics (Amdeberhan et al., 14 Aug 2025).