Tubdigons and Fine's Identity in Combinatorics

• Tubdigons are defined as roofed, subdivided polygons that generalize subdigons by including 2-gonal faces, serving as key objects in combinatorial proofs of Fine's identity.
• Their structure is recursively specified with unary and k-ary operators, linking them to plane trees and enabling a systematic combinatorial analysis.
• The enumeration through stars-and-bars and multinomial expansions directly leads to a combinatorial derivation of Fine's multinomial identity.

Searching arXiv for papers directly relevant to “Tubdigons,” including related work on subdigons and graph tubings. Tubdigons are roofed, subdivided polygons that generalize the subdigons of Wildberger and Rubine by allowing faces of degree $2$. In the formulation used to obtain a new proof of Fine’s identity, a tubdigon is a combinatorial and geometric object whose recursive structure yields a formal polynomial equation, and whose coefficients admit a second, direct stars-and-bars interpretation. Equating these two counts produces Fine’s multinomial identity (Rubine, 16 Aug 2025).

1. Definition and basic combinatorial structure

A subdigon of type

$\mathbf{m}=[m_2,m_3,m_4,\ldots]$

is a convex planar polygon with a distinguished side called the roof, subdivided by non-intersecting diagonals into smaller faces consisting of $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, and so on, with only finitely many $m_i$ nonzero. The null subdigon $|$ has $2$ vertices, $1$ edge, no faces, and type $[]$ (Rubine, 16 Aug 2025).

Tubdigons extend this class by allowing $\mathbf{m}=[m_2,m_3,m_4,\ldots]$0-gons. A tubdigon has type

$\mathbf{m}=[m_2,m_3,m_4,\ldots]$1

where $\mathbf{m}=[m_2,m_3,m_4,\ldots]$2 is the number of $\mathbf{m}=[m_2,m_3,m_4,\ldots]$3-gons, $\mathbf{m}=[m_2,m_3,m_4,\ldots]$4 the number of triangles, $\mathbf{m}=[m_2,m_3,m_4,\ldots]$5 the number of quadrilaterals, and so forth. Geometrically, $\mathbf{m}=[m_2,m_3,m_4,\ldots]$6-gons cannot be drawn with two straight edges in the plane, so at least one of the edges of every $\mathbf{m}=[m_2,m_3,m_4,\ldots]$7-gon is drawn curved (Rubine, 16 Aug 2025).

For a subdigon of type $\mathbf{m}=[m_2,m_3,m_4,\ldots]$8, the numbers of vertices, edges, and faces are

$\mathbf{m}=[m_2,m_3,m_4,\ldots]$9

and they satisfy Euler’s formula $m_2$0. A tubdigon keeps the same vertex count as its underlying subdigon and increases edges and faces by $m_2$1; equivalently, each added $m_2$2-gon contributes one new edge and one new face, but no new vertices (Rubine, 16 Aug 2025).

This generalization has a tree-theoretic interpretation. Subdigons are in natural bijection with plane trees with no unary nodes, whereas tubdigons include unary nodes and therefore correspond naturally to general plane trees (Rubine, 16 Aug 2025).

2. Recursive specification

Subdigons are built recursively from roofed central polygons. There is a family of $m_2$3-ary operators

$m_2$4

where $m_2$5 is formed by taking a central roofed $m_2$6-gon and attaching the subdigons $m_2$7 along the sides adjacent to the roof, in counterclockwise order. Every non-null subdigon is uniquely of the form $m_2$8 for some $m_2$9, which yields the structural equation

$m_3$0

for the multiset $m_3$1 of all subdigons (Rubine, 16 Aug 2025).

Tubdigons add a unary operator $m_3$2, corresponding to attaching a single tubdigon to a roofed $m_3$3-gon. If $m_3$4 denotes the multiset of all tubdigons, then every non-null tubdigon is exactly one of

$m_3$5

and the multiset specification becomes

$m_3$6

The addition of the unary operator is the precise combinatorial mechanism by which $m_3$7-gons enter the theory (Rubine, 16 Aug 2025).

The type data are encoded by an accounting monomial. For a subdigon $m_3$8 of type $m_3$9,

$m_4$0

while for a tubdigon $m_4$1 of type $m_4$2,

$m_4$3

If $m_4$4, then the generating function for all tubdigons is

$m_4$5

where $m_4$6 is the number of tubdigons of type $m_4$7 (Rubine, 16 Aug 2025).

3. Polynomial equation and Wildberger’s formula

Applying $m_4$8 to the tubdigon specification yields the functional equation

$m_4$9

or equivalently

$m_i$0

This is a polynomial equation of the form

$m_i$1

with

$m_i$2

(Rubine, 16 Aug 2025).

Wildberger and Rubine’s soft polynomial formula gives the formal series solution of such an equation. In the notation used for subdigons,

$m_i$3

satisfies

$m_i$4

and the soft polynomial formula expresses the solution of a general polynomial equation in terms of these coefficients $m_i$5 (Rubine, 16 Aug 2025).

Substituting the tubdigon coefficients into that formula gives

$m_i$6

The factor $m_i$7 is the formal contribution of the unary operator $m_i$8, and it is precisely the term whose multinomial expansion later produces Fine’s identity (Rubine, 16 Aug 2025).

4. Enumeration by type

There is also a direct combinatorial count of tubdigons of fixed type. Fix a subdigon $m_i$9 of type $|$0, with $|$1 edges. To obtain a tubdigon of type $|$2 lying over $|$3, one distributes $|$4 indistinguishable extra edge slots among the $|$5 distinguishable edges of $|$6. This is a stars-and-bars problem, so the number of such tubdigons over $|$7 is

$|$8

If $|$9 is the number of subdigons of type $2$0, then

$2$1

(Rubine, 16 Aug 2025).

The same source also records a closed form derived from Erdélyi and Etherington: $2$2 which yields the corresponding closed expression for $2$3. However, the stars-and-bars formula is the decisive ingredient for the proof of Fine’s identity (Rubine, 16 Aug 2025).

This enumeration situates tubdigons in an established combinatorial lineage. The count is described as well known in combinatorics, with related appearances in work of Raney on well-formed expressions of $2$4-ary function applications, Tutte on plane trees of a given type, and Kreweras on noncrossing partitions with and without singletons (Rubine, 16 Aug 2025).

5. Fine’s identity from coefficient extraction

The tubdigon proof of Fine’s identity compares the direct stars-and-bars count with the coefficient obtained from the polynomial solution. Expanding

$2$5

by the multinomial theorem gives

$2$6

After reindexing, this becomes

$2$7

(Rubine, 16 Aug 2025).

The direct count gives

$2$8

Setting

$2$9

one obtains

$1$0

which is Fine’s identity: $1$1 (Rubine, 16 Aug 2025).

In the tubdigon interpretation, $1$2 is the number of edges in the underlying subdigon, $1$3 is the number of added $1$4-gons, and the multinomial coefficient records how multiplicities are assigned to the base edges. The binomial coefficient on the right is the stars-and-bars count for placing those extra $1$5-gons among the $1$6 edges (Rubine, 16 Aug 2025).

6. Related notions and terminological boundaries

Tubdigons should be distinguished from two nearby but separate notions in the arXiv literature. In graph-associahedra, the basic object is a tubing on a finite simple graph: a non-empty family of tubes, containing the universal tube, such that every pair of tubes is compatible. Here a tube is a subset of nodes whose induced subgraph is connected, and the resulting poset of tubings realizes the graph-associahedron $1$7 (Forcey et al., 2019). This theory concerns graph combinatorics, substitution operations, and operadic categories rather than roofed subdivided polygons.

They should also be distinguished from digons in arrangements of pseudocircles. In that setting, a digon is a $1$8-cell in the cell structure of a simple arrangement of pairwise intersecting pseudocircles, and $1$9 denotes the number of digons in an arrangement $[]$0 (Felsner et al., 2022). The source explicitly states that the term “Tubdigons” does not appear there; the closest relevant concept is digons (Felsner et al., 2022).

The term therefore has a specific meaning in the Fine-identity context: a tubdigon is a roofed, subdivided polygon with possible $[]$1-gonal faces, positioned conceptually between subdigons and general plane trees, and technically between a recursive multiset specification and a formal polynomial solution (Rubine, 16 Aug 2025).