OEIS Sequence A393920 Overview

• OEIS Sequence A393920 is an integer sequence defined as Rₙ, emerging from the study of extension closed additive subcategories in uniformly oriented Aₙ-quivers.
• The research develops interlaced recurrences using two-parameter arrays and reveals a state-quotient graph whose vertex counts follow Fibonacci number patterns.
• The work also establishes exponential growth bounds and a finite lattice structure, integrating representation theory with combinatorial and topological insights.

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 (Mazorchuk, 1 Jul 2026).

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

$A_n$0

The prescribed initial conditions are

$A_n$1

with all other boundary values equal to zero.

For $A_n$2, the defining interlaced recurrences are

$A_n$3

$A_n$4

and

$A_n$5

The proof outline identifies these same recurrences with combinatorial quantities

$A_n$6

and similarly for $A_n$7. Summing over $A_n$8 yields $A_n$9 and $a(n,k)$0, and matching boundary values gives $a(n,k)$1 and $a(n,k)$2, hence $a(n,k)$3 (Mazorchuk, 1 Jul 2026).

2. Initial values and computational form

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

$a(n,k)$4	$a(n,k)$5
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 (Mazorchuk, 1 Jul 2026).

The recursive presentation is not a single scalar recurrence in $a(n,k)$6 alone. Instead, $a(n,k)$7 is recovered by summing over the auxiliary array $a(n,k)$8. 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 $a(n,k)$9, whose vertices at level $b(n,k)$0 are certain equivalence classes of partial data of elements of $b(n,k)$1. The number of vertices at level $b(n,k)$2 is

$b(n,k)$3

where

$b(n,k)$4

Equivalently, the ordinary generating function is

$b(n,k)$5

The same construction also yields the path-counting interpretation

$b(n,k)$6

Accordingly, $b(n,k)$7 can be computed in time roughly proportional to $b(n,k)$8 (Mazorchuk, 1 Jul 2026).

The Fibonacci connection is therefore not an identification of $b(n,k)$9 itself with a Fibonacci subsequence. Rather, the Fibonacci numbers control the size of the level sets of the state-quotient graph, while $R_n$0 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$1 and $R_n$2.

First, for naturally labeled posets, an interval $R_n$3 is identified with the point $R_n$4. Under this bijection, Condition $R_n$5 exactly encodes transitivity: if $R_n$6 and $R_n$7 overlap or touch, then their convex hull also lies in the set. This yields a bijection between $R_n$8 and the set of naturally labeled partial orders on $R_n$9 points (Mazorchuk, 1 Jul 2026).

Second, for Catalan objects, the subset

$a(n,k)$0

satisfies

$a(n,k)$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 $a(n,k)$2-set is convex if and only if it is generated by finitely many intervals. Since every convex topology contains the whole set $a(n,k)$3, the paper obtains a bijection

$a(n,k)$4

that is, with $a(n,k)$5. Hence

$a(n,k)$6

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 $a(n,k)$7 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 $a(n,k)$8.

For the lower bound, truncation of the recurrence together with checking initial conditions gives, for all $a(n,k)$9,

$b(n,k)$0

For the upper bound, weighted sums

$b(n,k)$1

and related quantities are introduced. From these, the paper derives a two-variable generating-function system whose dominant singularity occurs at

$b(n,k)$2

so that

$b(n,k)$3

Since $b(n,k)$4, it follows that

$b(n,k)$5

Together these yield

$b(n,k)$6

The paper also records numerical evidence that $b(n,k)$7 converges to about $b(n,k)$8, consistent with these bounds (Mazorchuk, 1 Jul 2026). 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, $b(n,k)$9 is a finite lattice. The paper gives an explicit description of several distinguished classes of elements.

The atoms are the singletons $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$0 for $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$1, and the join-irreducible elements are exactly the atoms.

The coatoms are the sets

$$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$2

for a total of $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$3.

The meet-irreducible elements are the complements of “axis-anchored” rectangles,

$$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$4

and

$$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$5

with

$$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$6

There are $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$7 such elements.

At small $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$8, the paper notes that these elements can be listed explicitly, and in particular the lattice is atomic and coatomic but not distributive in general (Mazorchuk, 1 Jul 2026). 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 $$A(n)=\sum_{k\in\mathbb Z} a(n,k), \qquad B(n)=\sum_{k\in\mathbb Z} b(n,k).$$9 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 $R_n=A(n), \qquad P_n=B(n),$0 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.