bibkey: mazorchukxu2026cokernel
authors: Volodymyr Mazorchuk, Shuoyang Xu
year: 2026
title: "On the number of cokernel-closed additive subcategories for uniformly oriented A_n quivers"
doi: 10.48550/arXiv.2610.07999
url: https://arxiv.org/html/2610.07999v2
claim: "Section 5.7 conjectures that every fixed first-column deficit i gives a polynomial of exact degree 2i and leading coefficient 1/i!."
strata_touched:
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuClosure
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuFamilies
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuOrder
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuGeneric
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuPatterns
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuReconstruction
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuNormalization
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuPolynomial
  - D5/S3/Combinatorics/BinomialBases/MazorchukXuTopPatterns
license: citation-only
triage: anchor

Mazorchuk and Xu: first-column deficit counts

Verified locator

DOI: 10.48550/arXiv.2610.07999

URL: https://arxiv.org/html/2610.07999v2

The source is version 2. Section 3, before Section 3.1, defines the triangular subsets and Condition (star), Section 5.1 defines the first-column statistic, and Section 5.7 states:

It seems reasonable to conjecture that $r(n,n+1-i)$ is a polynomial in $n$ of degree $2i$ and leading coefficient $\frac{1}{i!}$.

The introductory fixed-parameter context is “any fixed number of missing points.” Propositions 11–14 already give the deficits zero through three. Proposition 15 concerns the separate restriction that all rows be nonempty.

The versioned HTML has SHA-256 b0cb87d1bf47f986b282c2594bec44b78f65982e8f82ae227f5107eafe3468b1.

Literal finite model

For $n\in \mathbb{N}$, let $T(n)=\{(a,b):0\le a\le b\le n\}$. A subset $Y$ belongs to $R(n)$ if every three selected points $(a,b),(c,d),(e,f)$, allowing repetitions, with $a\le c,e\le b\le d,f$ force $(\max (c,e),\max (d,f))\in Y$ and, when $b<\min (d,f)$, $(b+1,\min (d,f))\in Y$. The statistic counts selected points with first coordinate zero. Empty rows and the empty subset are allowed.

The conjecture asks for one polynomial for each $i\in \mathbb{N}$, valid at every $n\in \mathbb{N}$ satisfying $i\le n+1$, with exact degree $2i$ and leading coefficient $1/i!$. It does not assert integer coefficients. Rational polynomials provide characteristic-zero numerical polynomials by the usual embedding into the reals. There is no value of $r(-1,k)$ in this domain and no field or representation-realization hypothesis in this combinatorial statement.

Mathematical scope

The point map $(a,b)\mapsto (a,b+1)$ is a counted equivalence with strict ordered edges on $0,\dots ,n+1$. The repeated-point condition becomes two universal edge rules. Monotone clique-fiber expansion preserves these rules; an increasing section gives the converse. Exceptional-to-ordinary incidence has a unique cutoff, including an empty ordinary list, and monotone threshold graphs satisfy both rules. Arbitrary cutoff data admit a bounded threshold completion; fully generic data have a unique completion. Sorting the actual anchor-neighbor set and its complement extracts position and cutoff data from every valid member of the literal finite family, with precisely i exceptions and s(j)=p(j)+j+1. Agreement of the anchor edges, cutoffs and exceptional-edge bits determines the entire labelled graph.

The finite weak-coordinate reconstruction gives an exact bijection with the literal source family. Grouping positions and cutoffs by their positive support gives

$$r(n,n+1-i)=\sum _{k=0}^{2i}{a}_{i,k}\left(\genfrac{}{}{0ex}{}{n+1-i}{k}\right).$$

The coefficients count admissible data at ordinary size $k$ using every positive coordinate; zero, ties, and exceptional-edge bits are retained. At support size $2i$, all entries are positive and distinct, and the exceptional bits have a unique admissible completion. Choosing the ordered position values and permuting the complementary cutoff values gives $i!a_{i,2i}=(2i)!$, including $i=0$. The corresponding rational falling-factorial sum is one polynomial on every original valid rank. Its nonzero coefficient at degree $2i$ is $1/i!$. The Lean declaration MazorchukXu.mazorchukXu_conjecture in D5/S3/Combinatorics/BinomialBases/MazorchukXuTopPatterns.lean proves the original Conjecture against the literal count, with no additional hypotheses.

No originality or priority claim is made. Bounded searches authenticated no exact existing Lean supplier; differently expressed equivalents and the full later literature are not excluded.