bibkey: daugherty2026counting
authors: Spencer Daugherty, Michaela Fahnestock, Shawn Huxtable, Michael Larson, Nourah Al Munayes and Storey Peacock
year: 2026
title: "Counting and Constructing Latin tableaux"
doi: null
url: https://arxiv.org/abs/2610.29291v1
claim: "Conjecture 6.3 states that for every integer k >= 5, the number LT(lambda) of Latin tableaux of shape lambda = (k,k-1,...,5,4,3,3) and conjugate type lambda-prime is 9*4^(k-3)."
strata_touched:
  - D5/S3/Combinatorics/Latin/Tableaux/Source
  - D5/S3/Combinatorics/Latin/Tableaux/MatchingTemplate
  - D5/S3/Combinatorics/Latin/Tableaux/Counting
license: citation-only
triage: anchor

Counting and Constructing Latin tableaux

Verified locator

URL: https://arxiv.org/abs/2610.29291v1

Version: arXiv:2610.29291v1, submitted 2026-10-21. The versioned author source and HTML Section 6 identify Conjecture 6.3 in “Open problems”. The inspected author TeX has SHA-256 328ad2a75c85bc166fad151ebd2515f9ba8f42c18bd56df618d3441237d52697. Its lines 739–741 state:

If $\lambda =(k,k-1,\dots ,5,4,3,3)$ with $k\ge 5$, then $LT(\lambda )=9\cdot 4^{k-3}$.

The author spelling above follows the versioned paper’s “Fahnestock”; abstract metadata spells that name “Fahnesock”. The TeX and HTML were inspected in the supplied source qualification; PDF rendering was not checked. No DOI is established here. citation-only describes this note’s use of the source, not a license attestation for the paper.

Counted objects and scope

Section 2 uses fixed left-aligned partition diagrams, with conjugation interchanging rows and columns. Definition 2.5 specifies the type by symbol multiplicities and prohibits an integer repeating in any row or column. Section 4 defines $LT(\lambda )$ using type ${\lambda }^{\prime }$. The source explains that each row then contains the symbols from 1 through its row length. These are full fillings of fixed cells with fixed integer symbols; the separately defined reduced counts are not $LT$. Equal-length bottom rows remain distinct positions. The endpoint $k=5$ has shape $(5,4,3,3)$.

The target has no finite upper cutoff or additional wide-partition hypothesis. Conjectures 6.1, 6.2 and the broader wide partition conjecture are separate assertions. A formal row-permutation encoding requires its own two-way correspondence with the source shape/type definition; the source observation is not a Lean proof of that bridge.

Open-status attestation and evidence limits

The paragraph preceding Conjecture 6.3 says that for this family with $k\ge 6$ nothing is known to be Latin by any result, although a simple counting formula appears available. This is the paper version’s open-status attestation, not worldwide certification. The nearby computation-through-size-22 statement concerns Conjectures 6.1 and 6.2; it is not adopted as evidence for Conjecture 6.3.

The supplied bounded prior-answer screen inspected returned arXiv metadata for exact Latin tableau/tableaux phrases and selected passages in 2007.13835v3, math/0205288v2, 2408.04086v1 and 2511.10548v1. No complete answer to this exact universal count was located in that inspected scope. Unindexed literature, equivalent earlier theorems, post-v1 answers and global priority remain unverified. A complete prior answer must be reused with zero new solution credit.

Issue 20269 registers the complete source assertion. The mathematical content is in Latin/Tableaux/Source, MatchingTemplate, and Counting under D5/S3/Combinatorics, with corresponding owning Blueprint Scribes. Counting.conjecture63_integer proves the entire integer-parametrized count for the literal unrestricted integer-filling definition. The canonical problem dossier explains the exact source correspondence, kernel-checked complete base, all-parameter invariant and proof boundaries. The source finiteness and finite enumeration are connected by explicit inverse maps, without a symmetry quotient. The ordinary counting argument and its independent source audit are admitted inputs; the Lean proof supplies distinct kernel evidence.

The proof begins at (5,4,3,3); the smaller (4,3,3) layer cannot support a uniform fourfold step. The four-choice invariant, rather than finite experimental evidence, supplies every subsequent transition. This note makes no CI, integration, publication or solved-problem credit claim. Current per-theorem audit evidence and any linked gaps must be reported separately. The bounded prior-art qualification above does not establish worldwide priority.