Let F_n^ind(sigma_1,…,sigma_k) be the set of indecomposable Fishburn permutations of length n avoiding the listed patterns, and let S_n(tau_1,…,tau_k) be the ordinary pattern-avoiding permutations in S_n. Fine-sequence conjecture. For every n≥ 1, |F_n^ind(2413,2431)|=|F_n^ind(2431,3241)|=|S_n-1(2413,3412,2143)|. The source reports verification for n≤ 15, identifies the common sequence as A033321, and notes that this conjecture is equivalent to the Catalan-convolution conjecture via the stated generating-function relation; it remains open.

Fine-sequence conjecture for indecomposable Fishburn classes

About 6 years old · traced to 2020

Let $F_n^{\mathrm{ind}}(\sigma_1,\ldots,\sigma_k)$ be the set of indecomposable Fishburn permutations of length $n$ avoiding the listed patterns, and let $S_n(\tau_1,\ldots,\tau_k)$ be the ordinary pattern-avoiding permutations in $S_n$ .

Fine-sequence conjecture. For every $n\geq 1$ ,

|F_n^{\mathrm{ind}}(2413,2431)|=|F_n^{\mathrm{ind}}(2431,3241)|=|S_{n-1}(2413,3412,2143)|.

The source reports verification for $n\leq 15$ , identifies the common sequence as A033321, and notes that this conjecture is equivalent to the Catalan-convolution conjecture via the stated generating-function relation; it remains open.

References

Primary source

Eric S. Egge, “Pattern-Avoiding Fishburn Permutations and Ascent Sequences”, arXiv:2208.01484 (2022).

Additional references

2 papers in this index state this conjecture (2020–2022). The statement above is taken from the most recent of them; the others are arXiv:2008.12297.

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

An unverified posted computation claims the printed equality fails at length six, while the published source reports agreement through length fifteen; neither the alleged error nor a correction has been independently verified.

Egge’s 2022 source records the conjecture that two indecomposable Fishburn classes and an ordinary pattern-avoiding class have equal counts for every $n\geq 1$ . It also relates the conjecture to a Catalan-convolution conjecture through generating functions.

Known results

The three sequences were computationally verified to agree for $n\leq 15$ and identified with OEIS A033321 (Egge, 2022).

Posted attempt

A posted exhaustive computation claims that at $n=6$ both Fishburn counts are $79$ , whereas the printed ordinary count is $77$ , and suggests replacing $3412$ by $3142$ ; it claims a likely typo, not a proof of the corrected conjecture. This attempt has not been independently verified.

Current status (as of August 2026): The conjecture remains open, with published verification only through $n\leq 15$ ; the alleged $n=6$ counterexample and proposed pattern correction are unconfirmed.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

Counterexample at n=6 and likely pattern transposition

The displayed equality is false as written. At $n=6$ , exhaustive evaluation of the definitions gives

|F_6^{\mathrm{ind}}(2413,2431)|=79

and

|F_6^{\mathrm{ind}}(2431,3241)|=79,

whereas

|S_5(2413,3412,2143)|=77.

For the first Fishburn class, splitting by first entry $3,4,5,6$ gives the counts $1,6,21,51$ , totaling $79$ . For the second class, the corresponding counts are $5,9,14,51$ , also totaling $79$ .

The ordinary count has a short inclusion-exclusion check. Among the 120 permutations in $S_5$ , the sets containing $2413$ , $3412$ , and $2143$ each have size 17. Their pairwise intersections have sizes $4,4,0$ , and the triple intersection is empty. Hence

120-(17+17+17)+(4+4+0)=77.

Thus the literal conjecture contains the false equality $79=77$ .

There is strong evidence for a transposition typo. Immediately after Conjecture 10.16, the source identifies the common sequence as OEIS A033321. That OEIS entry lists ordinary permutations avoiding

2413,\ 3142,\ 2143,

not $2413,3412,2143$ .

Replacing $3412$ by $3142$ makes the $S_5$ count equal to 79. All four resulting sequences also agree through the exhaustively checked range $n\leq 7$ .

This refutes the printed statement and identifies a likely correction; it does not prove the corrected equality for every $n$ .

Sources:

https://arxiv.org/abs/2208.01484

https://oeis.org/A033321