---
title: OEIS Sequence A393920 Overview
url: https://www.emergentmind.com/topics/oeis-sequence-a393920
type: topic
---

# OEIS Sequence A393920 Overview

Searching arXiv for the specified paper to ground the article in the cited source.
OEIS sequence A393920 is the integer sequence \(a_n=R_n\) studied in connection with the number of extension closed additive subcategories for uniformly oriented \(A_n\)-quivers. In the formulation developed by Mazorchuk, the sequence is obtained from two finitely supported two-parameter arrays \(a(n,k)\) and \(b(n,k)\), and the paper establishes a recurrence for \(R_n\), a connection with Fibonacci numbers, exponential lower and upper bounds, several combinatorial bijections, and a lattice-theoretic description of the underlying representation-theoretic poset [2607.00651].

## 1. Recursive definition

The paper introduces two two-parameter arrays \(a(n,k)\) and \(b(n,k)\), both supported on finite triangular regions, and then defines
\[
A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).
\]
Its main theorem states that
\[
R_n=A(n), \qquad P_n=B(n),
\]
and hence
\[
a_n=\mathrm{A393920}(n)=R_n=A(n).
\]

The arrays are extended by zero outside the regions
\[
\{(n,k)\mid n\ge -2,\;0\le k\le n+1\}
\quad\text{and}\quad
\{(n,k)\mid n\ge0,\;1\le k\le n+1\}.
\]
The prescribed initial conditions are
\[
a(-2,0)=a(-1,0)=a(0,0)=a(0,1)=1, \qquad b(0,1)=1,
\]
with all other boundary values equal to zero.

For \(n>0\), the defining interlaced recurrences are
\[
a(n,0)=\sum_{i=0}^{n} a(n-1,i),
\]
\[
a(n,k)=b(n,k)+\sum_{j=0}^{n-1} b(j,k)\sum_{i=0}^{n-j-1} a(n-j-2,i), \qquad k\ge 1,
\]
and
\[
b(n,k)=a(n-1,k-1)+\sum_{m=1}^{n}\sum_{r=0}^{\min(m-1,k-1)} a(m-2,r)\sum_{q=k-1-r}^{n-m+1} b(n-m,q).
\]

The proof outline identifies these same recurrences with combinatorial quantities
\[
r(n,k)=\bigl|\{\;Y\subseteq\mathcal Q(n)\text{ satisfying }(\star)\text{ with }|Y\cap\{0\}\times\{-\}|=k\}\bigr|
\]
and similarly for \(p(n,k)\). Summing over \(k\) yields \(R_n\) and \(P_n\), and matching boundary values gives \(r(n,k)=a(n,k)\) and \(p(n,k)=b(n,k)\), hence \(R_n=A(n)\) [2607.00651].

## 2. Initial values and computational form

The first values obtained from the SageMath implementation are as follows.

| \(n\) | \(R_n\) |
|---|---:|
| 0 | 2 |
| 1 | 7 |
| 2 | 34 |
| 3 | 199 |
| 4 | 1308 |
| 5 | 9300 |
| 6 | 69978 |
| 7 | 549559 |
| 8 | 4462570 |
| 9 | 37223311 |
| 10 | 317405288 |
| 11 | 2756819108 |
| 12 | 24321036896 |

These values agree with OEIS A393920. The paper states that further terms are readily obtained [2607.00651].

The recursive presentation is not a single scalar recurrence in \(R_n\) alone. Instead, \(R_n\) is recovered by summing over the auxiliary array \(a(n,k)\). This suggests that the sequence is governed by a richer state decomposition than is visible at the level of the one-dimensional sequence itself.

## 3. Fibonacci-state graph connection

A central structural result is the construction of a “state-quotient” graph \(\Sigma\), whose vertices at level \(n\) are certain equivalence classes of partial data of elements of \(\mathcal R(n)\). The number of vertices at level \(n\) is
\[
|\Sigma_n|=G_{n+1},
\]
where
\[
G_n=F_{2n+1},\qquad G_0=1,\;G_1=2,\qquad G_n=3G_{n-1}-G_{n-2}\quad(n\ge 2).
\]
Equivalently, the ordinary generating function is
\[
\sum_{n\ge 0} G_n z^n=\frac{1+z}{1-3z+z^2}.
\]

The same construction also yields the path-counting interpretation
\[
\text{the total number of directed root-to-level-}n\text{ paths in }\Sigma \text{ is exactly } R_n.
\]
Accordingly, \(R_n\) can be computed in time roughly proportional to \(F_{2n+3}\) [2607.00651].

The Fibonacci connection is therefore not an identification of \(R_n\) itself with a Fibonacci subsequence. Rather, the Fibonacci numbers control the size of the level sets of the state-quotient graph, while \(R_n\) counts root-to-level paths in that graph. This is the sense in which the paper describes the relation as “surprising.”

## 4. Combinatorial correspondences

The paper develops several bijective interpretations of the sets underlying \(R_n\) and \(P_n\).

First, for naturally labeled posets, an interval \([a,b]\subseteq\{1,\dots,n+1\}\) is identified with the point \((a-1,n-b)\in\mathcal Q(n)\). Under this bijection, Condition \((\star)\) exactly encodes transitivity: if \([a,b]\) and \([c,d]\) overlap or touch, then their convex hull also lies in the set. This yields a bijection between \(\mathcal R(n-2)\) and the set of naturally labeled partial orders on \(n\) points [2607.00651].

Second, for Catalan objects, the subset
\[
\mathcal R'(n)=\{Y\in\mathcal R(n)\mid \text{bottom row is full}\}
\]
satisfies
\[
|\mathcal R'(n)|=C_{n+1}=\frac{1}{n+2}\binom{2n+2}{n+1}.
\]
The paper derives this by a simple “one-row-split” argument recovering the usual Catalan recurrence.

Third, for convex topologies, a topology on a totally ordered \((n+1)\)-set is convex if and only if it is generated by finitely many intervals. Since every convex topology contains the whole set \([1,n+1]\), the paper obtains a bijection
\[
\{\text{convex topologies on }[1,n+1]\}\longleftrightarrow \{\,Y\in\mathcal R(n)\mid (0,0)\in Y\},
\]
that is, with \(\mathcal P(n)\). Hence
\[
P_n=B(n)
\]
is OEIS A234268.

These correspondences place A393920 at an intersection of representation theory, finite posets, Catalan combinatorics, and finite topological structures. A plausible implication is that the recursive complexity of \(R_n\) reflects a common closure phenomenon visible in each of these models.

## 5. Exponential bounds and asymptotic evidence

The paper proves both lower and upper exponential bounds for the growth of \(R_n\).

For the lower bound, truncation of the recurrence together with checking initial conditions gives, for all \(n\),
\[
R_n\ge \frac{1}{12}\,9^n.
\]

For the upper bound, weighted sums
\[
U_n=\sum_k \Bigl(\frac32\Bigr)^k a(n-1,k)
\]
and related quantities are introduced. From these, the paper derives a two-variable generating-function system whose dominant singularity occurs at
\[
\rho=\frac{7-\sqrt{33}}{16},
\]
so that
\[
U_n=O(\rho^{-n})=O\bigl((7+\sqrt{33})^n\bigr).
\]
Since \(R_n\le U_{n+1}\), it follows that
\[
R_n<(7+\sqrt{33}+\varepsilon)^n \qquad (\forall \varepsilon>0,\; n\gg 0).
\]

Together these yield
\[
\frac{1}{12}\,9^n\le R_n<(7+\sqrt{33}+\varepsilon)^n\approx (12.7446)^n.
\]

The paper also records numerical evidence that \(R_{n+1}/R_n\) converges to about \(10.5\), consistent with these bounds [2607.00651]. This suggests an exponential growth rate strictly between the proven lower and upper estimates, although no sharper asymptotic constant is stated in the summary.

## 6. Lattice structure and related subsequences

Ordered by inclusion, \(\mathcal R(n)\) is a finite lattice. The paper gives an explicit description of several distinguished classes of elements.

The atoms are the singletons \(\{(x,y)\}\) for \((x,y)\in\mathcal Q(n)\), and the join-irreducible elements are exactly the atoms.

The coatoms are the sets
\[
\mathcal Q(n)\setminus\bigl\{(k,0),(k,1),\dots,(k,n-k)\bigr\},
\qquad
\mathcal Q(n)\setminus\bigl\{(0,k),(1,k),\dots,(n-k,k)\bigr\}
\quad (k=1,\dots,n),
\]
for a total of \(2n\).

The meet-irreducible elements are the complements of “axis-anchored” rectangles,
\[
\mathcal Q(n)\setminus\{(x,y)\mid a\le x\le b,\;0\le y\le n-b\},
\]
and
\[
\mathcal Q(n)\setminus\{(x,y)\mid 0\le x\le n-d,\;c\le y\le d\},
\]
with
\[
1\le a\le b\le n,\qquad 0\le c\le d\le n.
\]
There are \((n+1)^2\) such elements.

At small \(n\), the paper notes that these elements can be listed explicitly, and in particular the lattice is atomic and coatomic but not distributive in general [2607.00651]. This separates the lattice from more rigid distributive frameworks often encountered in order-theoretic enumeration.

The summary also records two further structural remarks. One is a subsequence of \(\mathcal R(n)\) in which the bottom row consists of the odd-indexed points; this coincides with OEIS A137842 via another “hook-split” and yields a new three-step path model. The other is that the state-graph \(\Sigma\) is simple and has a single outgoing edge-label at each vertex; this underlies the fast algorithm that Copilot first discovered.

Taken together, these results present A393920 as more than a numerical sequence. It is the enumerative shadow of a finite lattice with explicit irreducible structure, of a state-graph with Fibonacci-governed level sizes, and of several bijectively equivalent combinatorial classes.

Source: https://www.emergentmind.com/topics/oeis-sequence-a393920