The 1243 inversion generating-function conjecture for rectangular posets

From papers

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for linear extensions of the rectangular poset with parameters $s,t$ that avoid the permutation pattern $\sigma$ , and let $[n]_q$ denote the $q$ -integer. The 1243 inversion generating-function conjecture. For all $t \ge 1$ ,

EN_{3,2t-1}(1243)(q) = q^{3(t^2-t+1)} [2t-1]_q [4t-1]_q.

The formula is a proposed $q$ -analogue for pattern-avoiding linear extensions; computer-generated data verify it for $t \le 9$ , while the general case remains open.

Progress summary

Solved

The published record shows that the formula is misstated, while the intended corrected formula has been proved.

The 2016 conjecture proposed an explicit inversion-generating formula for $1243$ -avoiding linear extensions of rectangular posets, supported computationally through $t\le 9$ . A later paper identifies the relevant statement as incorrectly parameterized and proves the corrected identity.

Known results

Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan (2016): the displayed $EN_{3,2t-1}(1243)(q)$ formula was recorded as Conjecture 7.1 and verified computationally for $t\le 9$ .
The same work established related univariate enumeration and inversion-number bounds for $1243$ -avoiding extensions.

2019 correction and proof

Defant’s Theorem 3.1 states and proves, for every $t\ge 1$ , the corrected formula for $EN_{3,t}(1243)(q)$ , with direct checks for $t\in\{1,2\}$ and an insertion classification for $t\ge 3$ . Thus the literal $EN_{3,2t-1}$ formulation is an erroneous version of a theorem that is now proved.

Current status (as of August 2026): the stated $EN_{3,2t-1}(1243)(q)$ conjecture is false as written, while the intended corrected $EN_{3,t}(1243)(q)$ identity is proved in the published literature.

Sources

Sources & referencesView supporting material

Primary source

David Anderson, Eric S. Egge, Manda Riehl, Lucas Ryan, Ruth Steinke and Yuriko Vaughan, “Pattern Avoiding Linear Extensions of Rectangular Posets”, arXiv:1605.06825 (2016).

1 week ago

The statement as written is false already at $t=1$ .

For $t=1$ , the poset $EN_{3,1}$ is a three-element chain. Its unique linear extension is $321$ , which has three inversions and automatically avoids the length-four pattern $1243$ . Therefore

EN_{3,1}(1243)(q)=q^3.

The proposed right-hand side is instead

q^{3(1^2-1+1)}[1]_q[3]_q = q^3(1)(1+q+q^2) = q^3+q^4+q^5.

Hence

q^3\neq q^3+q^4+q^5,

which disproves the literal MathDB statement.

This is a known error in the original formulation. Colin Defant explicitly notes that it was stated incorrectly and proves the corrected result in Theorem 3.1 of “Proofs of Conjectures about Pattern-Avoiding Linear Extensions”:

https://arxiv.org/abs/1905.02309

Published version:

https://doi.org/10.23638/DMTCS-21-4-16

The corrected theorem is

EN_{3,t}(1243)(q) = \frac{ q^{3(t^2-t+1)} \left( 1-q^{2t-1}-2q^{2t}+q^{3t-1}+q^{3t} \right) }{ (1-q)(1-q^2) }

for every $t\geq1$ . Notice that the corrected left side is $EN_{3,t}$ , not $EN_{3,2t-1}$ .

For $t\geq3$ , Defant’s insertion classification gives the exact finite sum

EN_{3,t}(1243)(q) = \sum_{i=1}^{t}\sum_{j=i+1}^{2t} q^{3t^2-3t+i+j}.

Summing the two geometric progressions yields precisely the corrected rational expression above; the cases $t=1,2$ are checked directly in the paper.

Thus the MathDB statement is disproved at its first parameter, while the intended corrected theorem is proved in the published literature.

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Samuel Schlesinger · 4 days ago