Multivariate Hyper-Catalan Series: Exploration Summary

The Original Problem

Wildberger and Rubine (2025) showed that the geometric polynomial equation:

$1 - \alpha + t_2 \alpha^2 + t_3 \alpha^3 + \cdots = 0$

has the series solution:

$\alpha = S[t_2, t_3, \ldots] = \sum_{m \geq 0} C_m t^m$

where $C_m$ is the hyper-Catalan number counting subdigons (roofed polygons with subdivisions).

Proposed Multivariate Extension

The Natural System

For $n$ variables, the geometric form becomes a coupled system:

$1 - x_i + \sum_{|\mathbf{k}| \geq 2} t^{(i)}_{\mathbf{k}} \, x_1^{k_1} \cdots x_n^{k_n} = 0, \quad i = 1, \ldots, n$

Bivariate Case (Detailed Analysis)

For two variables $(x, y)$ with pure quadratic terms:

x &= 1 + a_{20} x^2 + a_{11} xy + a_{02} y^2 \\
y &= 1 + b_{20} x^2 + b_{11} xy + b_{02} y^2
\end{aligned}$$

The solutions are power series:

$$x = \sum C^{(x)}[\mathbf{m}, \mathbf{n}] \prod a_{jk}^{m_{jk}} \prod b_{jk}^{n_{jk}}$$

## Key Discoveries

### 1. Univariate Recovery

When all $b$-coefficients are zero, the pure $a_{20}^n$ terms recover exactly the Catalan numbers:

| Term | Coefficient | Catalan $C_n$ |
|------|-------------|---------------|
| $a_{20}^0$ | 1 | $C_0 = 1$ |
| $a_{20}^1$ | 1 | $C_1 = 1$ |
| $a_{20}^2$ | 2 | $C_2 = 2$ |
| $a_{20}^3$ | 5 | $C_3 = 5$ |
| $a_{20}^4$ | 14 | $C_4 = 14$ |
| $a_{20}^5$ | 42 | $C_5 = 42$ |

### 2. Combinatorial Interpretation: Colored Binary Trees

The coefficient of $\prod a_{jk}^{m_{jk}} \prod b_{jk}^{n_{jk}}$ in $x$ counts **2-colored plane binary trees** where:

- Each internal node has color **A** (x-type) or **B** (y-type)
- An A-node with coefficient $a_{jk}$ has $j$ X-children and $k$ Y-children
- A B-node with coefficient $b_{jk}$ has $j$ X-children and $k$ Y-children
- The tree is rooted at an X-type position

### 3. Tree Structure Constraints

For a valid X-rooted tree:
- **Total internal nodes**: $n = n_A + n_B$ where $n_A = \sum m_{jk}$, $n_B = \sum n_{jk}$
- **X-children**: $x_{ch} = 2m_{20} + m_{11} + 2n_{20} + n_{11}$
- **Y-children**: $y_{ch} = m_{11} + 2m_{02} + n_{11} + 2n_{02}$
- **Validity constraint**: $x_{ch} \geq n_A$ (enough X-positions for A-nodes plus root)
- **Validity constraint**: $y_{ch} \geq n_B$ (enough Y-positions for B-nodes)

### 4. Interesting Patterns

**Coefficient of $a_{20}^{n-1} \cdot a_{11}$:**
- $a_{20} \cdot a_{11}$: 3 = $\binom{3}{1}$
- $a_{20}^2 \cdot a_{11}$: 10 = $\binom{5}{2}$
- $a_{20}^3 \cdot a_{11}$: 35 = $\binom{7}{3}$
- $a_{20}^4 \cdot a_{11}$: 126 = $\binom{9}{4}$

Pattern: $\binom{2n-1}{n-1}$ for $n$ total internal nodes

**Zero coefficients reveal constraints:**
- $a_{20} \cdot b_{20}$: coefficient = 0
- Reason: If an A-node has 2 A-children, a B-node cannot exist in the same tree at degree 2

### 5. Partial Formula Match (Raney-style)

For certain terms, the formula:

$$C = \frac{x_{ch}}{2n} \cdot \frac{(2n)!}{\prod m_{jk}! \cdot \prod n_{jk}!}$$

matches the computed coefficients, specifically when $m_{11}$ and $a_{20}$ are both present with small $b$-degree.

## Conjectured Formula

Based on the analysis, the **multivariate hyper-Catalan number** should be:

$$C^{(x)}[\mathbf{m}, \mathbf{n}] = \frac{(E-1)!}{(V_x - 1)! \cdot V_y! \cdot \mathbf{m}! \cdot \mathbf{n}!} \cdot f(x_{ch}, y_{ch}, n_A, n_B)$$

where:
- $E = 2n$ (total edges in binary tree with $n$ nodes)
- $V_x = x_{ch} + 1$ (X-vertices including root)
- $V_y = y_{ch}$ (Y-vertices)
- $f$ is a correction factor related to the ballot problem

The exact form of $f$ requires further investigation, but should involve:
- The ballot lemma: counting sequences where X always leads
- The Good multivariate Lagrange inversion formula

## Connection to Known Structures

1. **Multivariate Lagrange-Good Inversion** (Good, 1960): Provides a general formula for extracting coefficients from systems of functional equations

2. **Colored Non-crossing Partitions**: The multivariate case should connect to partitions of multiple cycles

3. **Associahedra and Tamari Lattices**: Higher-dimensional generalizations

## Future Directions

1. **Derive the exact closed-form formula** using Good's multivariate Lagrange inversion
2. **Prove the combinatorial interpretation** via bijection with colored plane trees
3. **Investigate the Geode factorization** for the multivariate case
4. **Connect to representation theory** and root systems
5. **Explore applications** to solving multivariate polynomial systems numerically

## Sample Coefficient Table (Bivariate Quadratic)

| Monomial | Coefficient | $n_A$ | $n_B$ | $x_{ch}$ | $y_{ch}$ |
|----------|-------------|-------|-------|----------|----------|
| 1 | 1 | 0 | 0 | 0 | 0 |
| $a_{20}$ | 1 | 1 | 0 | 2 | 0 |
| $a_{11}$ | 1 | 1 | 0 | 1 | 1 |
| $a_{02}$ | 1 | 1 | 0 | 0 | 2 |
| $a_{20}^2$ | 2 | 2 | 0 | 4 | 0 |
| $a_{20} a_{11}$ | 3 | 2 | 0 | 3 | 1 |
| $a_{20} a_{02}$ | 2 | 2 | 0 | 2 | 2 |
| $a_{11}^2$ | 1 | 2 | 0 | 2 | 2 |
| $a_{20}^3$ | 5 | 3 | 0 | 6 | 0 |
| $a_{20}^4$ | 14 | 4 | 0 | 8 | 0 |
| $a_{02} b_{02}$ | 2 | 1 | 1 | 0 | 4 |
| $a_{02} b_{11}$ | 2 | 1 | 1 | 1 | 3 |
| $a_{11} b_{20}$ | 1 | 1 | 1 | 3 | 1 |

## References

1. Wildberger, N.J. & Rubine, D. (2025). "A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode." *American Mathematical Monthly*, 132(5), 383-402.

2. Good, I.J. (1960). "Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes." *Mathematical Proceedings of the Cambridge Philosophical Society*, 56(4), 367-380.

3. Raney, G.N. (1960). "Functional composition patterns and power series reversion." *Transactions of the American Mathematical Society*, 94, 441-451.