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

From papers

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.

Progress summary

Open

The conjecture matches the known sequence through length $15$ , but no proof or verified disproof has appeared.

The conjecture asserts that two indecomposable Fishburn-avoidance classes have the same size as a corresponding class of ordinary pattern-avoiding permutations for every $n\geq 1$ . The source also states that it is equivalent, through a generating-function relation, to the Catalan-convolution conjecture.

Known results

The three sequences were reported to agree for $n\leq 15$ and to equal OEIS sequence A033321.
The 2018 Fishburn-permutation paper recorded related indecomposable equinumerosity conjectures and explicitly left such questions open.

No verified resolution through August 2026

The retrieved literature contains no proof, counterexample, correction, referee report, or independent exposition resolving the stated conjecture. Related later work proves other Fishburn enumeration results but does not address this equality.

Current status (as of August 2026): The conjecture remains open; only the reported finite verification through $n\leq 15$ and its equivalence with the Catalan-convolution conjecture are settled.

Sources

Sources & referencesView supporting material

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.

1 week ago

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

0 endorsements

Samuel Schlesinger · 5 days ago