The 1243 inversion generating-function conjecture for rectangular posets

About 10 years old · traced to 2016

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.

References

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).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The original formula is false as written, and a published paper proves the corrected version; a posted calculation also claims a first-case counterexample but has not been independently verified.

The conjecture was recorded by Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan in 2016 as a proposed formula for inversion-generating functions of $1243$ -avoiding linear extensions of rectangular posets. Their computations supported it through $t\leq 9$ , but did not prove the general claim.

Known results

Anderson et al. (2016) proved the ordinary $1243$ -avoiding enumeration via Fuss–Catalan numbers and established inversion-number bounds.
The same paper explicitly left the relevant inversion distribution as an open investigation and stated the displayed identity only as Conjecture 7.1.

2019 correction and proof

Defant states that the conjecture was parameterized incorrectly and proves the corrected identity for $EN_{3,t}(1243)(q)$ , for every $t\geq 1$ , with direct checks for $t\in\{1,2\}$ and an insertion classification for $t\geq 3$ . Thus the literal $EN_{3,2t-1}$ claim is false, while the intended corrected theorem is proved.

Posted attempt

A posted calculation claims a complete disproof at $t=1$ , comparing $q^3$ with $q^3+q^4+q^5$ , and identifies the corrected parameterization. This attempt has not been independently verified.

Current status (as of August 2026): the stated $EN_{3,2t-1}(1243)(q)$ conjecture is false, and Defant’s corrected $EN_{3,t}(1243)(q)$ identity is proved in the literature.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.