Geode in Hyper-Catalan Series

• Geode is the reduced factor in hyper-Catalan series, defined by removing the linear sum from the generating function of geometric polynomial equations.
• It bridges combinatorial models like lattice paths and ordered trees with explicit recurrences and closed-form projections that generalize Catalan numbers.
• Its computational complexity escalates with dimensionality, shifting from closed forms in 2D to holonomic recurrences in higher dimensions.

Searching arXiv for papers on the mathematical Geode associated with hyper-Catalan series and related combinatorial interpretations. Geode is the reduced factor in the hyper-Catalan series solution of geometric polynomial equations. If $S$ denotes the hyper-Catalan generating series characterized by $1-S+t_2S^2+t_3S^3+\cdots=0$, then its positive part factors as $S-1=(t_2+t_3+t_4+\cdots)G$, and the quotient $G$ is called the Geode. In a parallel notation one writes $S=1+\sum_{n\ge 1} t_nS^n$ and $G=(S-1)/S_1$ with $S_1=\sum_{n\ge 1} t_n$. The Geode is therefore not a separate enumerative theory, but the factorized and layered descendant of the hyper-Catalan array; subsequent work connected it to lattice paths, ordered trees, explicit recurrences, and computational problems of rapidly increasing difficulty (Rubine, 17 Jul 2025, Rubine, 6 Jul 2025).

1. Definition and algebraic origin

The Geode enters through the hyper-Catalan generating sum

$$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$

where $C[m_2,m_3,m_4,\ldots]$ is the hyper-Catalan number counting roofed polygon subdivisions into $m_2$ triangles, $1-S+t_2S^2+t_3S^3+\cdots=0$0 quadrilaterals, $1-S+t_2S^2+t_3S^3+\cdots=0$1 pentagons, and so on. The series is a formal zero of

$1-S+t_2S^2+t_3S^3+\cdots=0$2

equivalently

$1-S+t_2S^2+t_3S^3+\cdots=0$3

Wildberger’s factorization

$1-S+t_2S^2+t_3S^3+\cdots=0$4

defines the Geode $1-S+t_2S^2+t_3S^3+\cdots=0$5 as the quotient obtained after removing the linear sum of the variables from the positive part of the hyper-Catalan series (Rubine, 6 Jul 2025).

Two closely related indexing conventions appear in the literature. In the polygon-dissection convention, the variables begin at $1-S+t_2S^2+t_3S^3+\cdots=0$6. In the ordered-tree convention, one instead writes

$1-S+t_2S^2+t_3S^3+\cdots=0$7

The second form is adapted to ordered trees of prescribed downdegree sequence, while the first is adapted to roofed polygons and subdigons. This suggests that the change of indexing reflects the underlying combinatorial model rather than a change of algebraic substance (Gossow, 24 Jul 2025).

2. Role in the series solution of polynomial equations

The Geode belongs to a broader formal-power-series method for solving polynomial equations. The starting point is the general geometric polynomial

$1-S+t_2S^2+t_3S^3+\cdots=0$8

whose formal root is

$1-S+t_2S^2+t_3S^3+\cdots=0$9

These coefficients are the hyper-Catalan numbers. In this framework, the Geode is the reduced hyper-Catalan array obtained by normalizing or projecting the same series solution rather than by introducing a new equation of a different type (Rubine, 17 Jul 2025).

A further refinement inserts an auxiliary face-counting parameter. If

$S-1=(t_2+t_3+t_4+\cdots)G$0

then

$S-1=(t_2+t_3+t_4+\cdots)G$1

This equation is formally analogous to the hyper-Catalan equation, but now $S-1=(t_2+t_3+t_4+\cdots)G$2 tracks face counts. The Geode viewpoint isolates the factor remaining after the obvious linear part is removed, so that the resulting coefficients form a layered refinement of the same polynomial-solving mechanism. This is why Geode-related arrays such as the Jumbo Geode arise naturally from the same formal root machinery (Rubine, 17 Jul 2025).

3. Recurrences and closed forms

A central structural result is the Geode recurrence derived from the factorization of $S-1=(t_2+t_3+t_4+\cdots)G$3. For a nonzero type vector $S-1=(t_2+t_3+t_4+\cdots)G$4, define its set of lessers by

$S-1=(t_2+t_3+t_4+\cdots)G$5

Then the coefficients satisfy the Lesser Geode Sum

$S-1=(t_2+t_3+t_4+\cdots)G$6

Equivalently, if $S-1=(t_2+t_3+t_4+\cdots)G$7 is any index $S-1=(t_2+t_3+t_4+\cdots)G$8 with $S-1=(t_2+t_3+t_4+\cdots)G$9, then

$G$0

This makes every Geode coefficient an integer combination of hyper-Catalan numbers and gives an effective recursive reduction from $G$1 to $G$2 (Rubine, 6 Jul 2025).

Several nontrivial coefficient families are known in closed form. If the type has only one nonzero component, $G$3, then

$G$4

For two consecutive shapes,

$G$5

where

$G$6

The bi-tri specialization is

$G$7

These formulas were posed as conjectures by Wildberger and later proved by recurrence and constant-term methods (Rubine, 6 Jul 2025, Amdeberhan et al., 22 Jun 2025).

The hyper-Catalan array itself also satisfies a multivariate recurrence obtained from $G$8, generalizing the classical Catalan convolution

$G$9

The Geode recurrence may therefore be viewed as a second-level convolution law induced by factoring the hyper-Catalan solution through its linear part (Rubine, 6 Jul 2025).

4. Combinatorial meaning and correction of an early conjecture

Early discussion treated the Geode combinatorics as unresolved. The exercises paper proposed, in the Bi-Tri case, that Geode coefficients might count ordered incomplete trees with $S=1+\sum_{n\ge 1} t_nS^n$0 binary nodes, $S=1+\sum_{n\ge 1} t_nS^n$1 ternary nodes, and so forth, together with a single additional leaf node. Sample comparisons already show that the Geode is strictly richer than the hyper-Catalan array: $S=1+\sum_{n\ge 1} t_nS^n$2 The same source also records that a naive geometric rule for the extra leaf fails, especially for $S=1+\sum_{n\ge 1} t_nS^n$3, so the conjectural picture was incomplete (Rubine, 17 Jul 2025).

A precise interpretation was then given in terms of weighted lattice paths. With step set $S=1+\sum_{n\ge 1} t_nS^n$4, weight $S=1+\sum_{n\ge 1} t_nS^n$5 for step $S=1+\sum_{n\ge 1} t_nS^n$6, and weight $S=1+\sum_{n\ge 1} t_nS^n$7 for step $S=1+\sum_{n\ge 1} t_nS^n$8, the series $S=1+\sum_{n\ge 1} t_nS^n$9 is the generating function for excursions, the Geode $G=(S-1)/S_1$0 is the generating function for nonnegative paths, and

$G=(S-1)/S_1$1

is the generating function for positive paths. The reciprocal identity

$G=(S-1)/S_1$2

makes coefficientwise nonnegativity immediate, and the proof is organized through free monoids and prime-path decompositions (Gessel, 12 Jul 2025).

An ordered-tree interpretation was established shortly afterward. If

$G=(S-1)/S_1$3

then $G=(S-1)/S_1$4 counts pairs $G=(S-1)/S_1$5 where $G=(S-1)/S_1$6 is an ordered tree of type $G=(S-1)/S_1$7 and $G=(S-1)/S_1$8 is a leaf visited before any non-leaf node in post-order traversal. Through a bijection with subdigons, external faces correspond to clawed nodes and external edges correspond to leaves. This corrected the earlier false conjecture that the Geode simply counts trees of the same type with one extra leaf; the traversal condition is essential (Gossow, 24 Jul 2025).

5. Projections, specializations, and named slices

The Geode supports many projections obtained by specializing variables or reorganizing coefficients into lower-dimensional slices. Some one-dimensional slices collapse to familiar sequences:

Slice	Identification
$G=(S-1)/S_1$9	Catalan numbers A000108
$S_1=\sum_{n\ge 1} t_n$0	Fuss numbers A001764
$S_1=\sum_{n\ge 1} t_n$1	A002293
$S_1=\sum_{n\ge 1} t_n$2	A002294

These identities show that the Geode interpolates between the full multivariate hyper-Catalan array and classical one-parameter Catalan-type sequences (Rubine, 6 Jul 2025).

Variable specializations recover standard lattice-path generating functions. When $S_1=\sum_{n\ge 1} t_n$3 and $S_1=\sum_{n\ge 1} t_n$4, the series collapses to the Catalan generating function. When $S_1=\sum_{n\ge 1} t_n$5 and $S_1=\sum_{n\ge 1} t_n$6, it yields Motzkin numbers and Riordan numbers. When $S_1=\sum_{n\ge 1} t_n$7 and $S_1=\sum_{n\ge 1} t_n$8, it yields large and small Schröder numbers. These specializations explain why Geode-related slices naturally produce Catalan, Riordan, and Schröder behavior within a single algebraic framework (Gessel, 12 Jul 2025).

The exercises paper also records Geode versions of classical projected arrays. The Little Schroeder Geode polynomial contains the segment

$S_1=\sum_{n\ge 1} t_n$9

The Riordan Geode polynomial contains

$$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$0

The Cayley Geode is given as a bivariate polynomial in $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$1 and $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$2, beginning with

$$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$3

These are not ad hoc analogues but projections of the same reduced hyper-Catalan object (Rubine, 17 Jul 2025).

6. Computation and dimensional complexity

Although the Geode is easy to define, its computation becomes difficult as the number of active indices grows. The direct approach is to build the hyper-Catalan polynomial $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$4, divide by $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$5, and extract coefficients. In two dimensions this is tractable because a closed form is available; the paper on computational complexity reports that $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$6 takes about $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$7 seconds by the original definition but about $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$8 seconds by the explicit formula. In three dimensions no general closed form is given, but the coefficients satisfy second-order linear recurrences with rational coefficients in each coordinate, and the diagonal sequence $$S=[t_2,t_3,t_4,\ldots] =\sum_{m_2,m_3,m_4,\ldots\ge 0} C[m_2,m_3,m_4,\ldots]\, t_2^{m_2}t_3^{m_3}t_4^{m_4}\cdots,$$9 satisfies a recurrence with rational functions $C[m_2,m_3,m_4,\ldots]$0 and $C[m_2,m_3,m_4,\ldots]$1 whose numerators and denominators are polynomials of degree $C[m_2,m_3,m_4,\ldots]$2. The initial values are

$C[m_2,m_3,m_4,\ldots]$3

and $C[m_2,m_3,m_4,\ldots]$4 is reported as computable in $C[m_2,m_3,m_4,\ldots]$5 seconds using the guessed recurrences (Amdeberhan et al., 14 Aug 2025).

Four dimensions created a sharper barrier. A later paper solved the diagonal challenge by using the coefficient identity

$C[m_2,m_3,m_4,\ldots]$6

together with slice-by-slice caches for both hyper-Catalan and Geode coefficients and multiplicative ratio updates for nearby hyper-Catalan entries. This method computed the exact $C[m_2,m_3,m_4,\ldots]$7-digit value of

$C[m_2,m_3,m_4,\ldots]$8

in about $C[m_2,m_3,m_4,\ldots]$9 hours (Rubine, 25 Dec 2025).

This computational profile suggests a pronounced dimensional transition: closed form in $m_2$0D, efficient holonomic-style recurrence in $m_2$1D, and substantially heavier exact arithmetic in $m_2$2D.

7. Later algebraic generalizations

The Geode also admits a noncommutative reformulation. Starting from the classical Lagrange series

$m_2$3

with $m_2$4, Wildberger and Rubine’s geode becomes

$m_2$5

In the noncommutative symmetric-function setting, the coefficients of $m_2$6 are polynomials in the $m_2$7 with nonnegative integer coefficients, and the series acquires interpretations in terms of Łukasiewicz words, Polish codes of plane rooted trees, parking functions, parking quasi-ribbons, and prime parking functions. One convenient operator description is

$m_2$8

independent of $m_2$9, which makes the quotient structure canonical in the noncommutative setting (Novelli et al., 23 Nov 2025).

This framework also yields $1-S+t_2S^2+t_3S^3+\cdots=0$00-geodes and $1-S+t_2S^2+t_3S^3+\cdots=0$01-geodes. The $1-S+t_2S^2+t_3S^3+\cdots=0$02-geodes arise from the $1-S+t_2S^2+t_3S^3+\cdots=0$03-Lagrange series

$1-S+t_2S^2+t_3S^3+\cdots=0$04

while the $1-S+t_2S^2+t_3S^3+\cdots=0$05-geodes arise from a more general hyper-Lagrange series

$1-S+t_2S^2+t_3S^3+\cdots=0$06

In these extensions the geode remains a quotient object with positive coefficients, but its combinatorics expands from ordered trees and lattice paths to Schröder trees and related Hopf-algebraic structures. This suggests that the original Geode is best understood not as an isolated array, but as the first member of a broader family of factorized Lagrange-series constructions (Novelli et al., 23 Nov 2025).