bibkey: elizalde2024pattern authors: Sergi Elizalde, Amya Luo year: 2024 title: “Pattern avoidance in nonnesting permutations” doi: 10.48550/arXiv.2412.00336 url: https://arxiv.org/abs/2412.00336v6 claim: “In Table 4 we list some cases that seem to give interesting enumeration sequences. All the conjectures have been checked for n up to 8.” strata_touched:

• D5/S3/Combinatorics/Nonnesting/NonnestingFour
• D5/S3/Combinatorics/Nonnesting/NonnestingOneThreeTwoTwo
• D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLow
• D5/S3/Combinatorics/Nonnesting/NonnestingRoyalHigh license: citation-only triage: anchor

Elizalde and Luo, pattern avoidance in nonnesting permutations

The paper enumerates nonnesting permutations of the multiset {1,1,2,2,…,n,n} avoiding every set of at least two patterns of length three and several sets of patterns of length four, and closes with a table of conjectured enumerations for further sets of patterns of length four.

Verified locator

DOI: 10.48550/arXiv.2412.00336

URL: https://arxiv.org/abs/2412.00336v6

• Locator: Section 1, a word contains a pattern when some subsequence is in the same relative order, equal letters included; nonnesting permutations are the permutations of {1,1,…,n,n} avoiding 1221 and 2112, and c_n(Λ) counts those avoiding every pattern of Λ.
• Locator: Section 4, Further research, Table 4: {1322} with (1/n) Σ_{k=0}^{n-1} C(3n,k) C(2n-k-2,n-1) (A007297); {1132,2213} and {1233,1322} with ordinary generating function ((1-x)^2 - √((1-x)^4 - 4x(1-x)^2))/(2x) (A006319); {1132,3312} with 3^n - 3·2^{n-1} + 1 (A168583); {1231,1312,2231,3221} with ordinary generating function (1-3x+2x^2)/((1-3x)(1-x-x^2)) (A099159).

Reading of the statement

For Λ = {1231,1312,2231,3221} the conjectured counts for n = 0, 1, …, 9 are 1, 1, 4, 11, 33, 98, 293, 877, 2628, 7879, the coefficients of the displayed rational function.

Bounded prior-resolution evidence

Read on 2026-09-30: the Semantic Scholar citation list of arXiv:2412.00336 contains arXiv:2502.13309 (Archer and Laudone, one pattern of length three in noncrossing and nonnesting permutations), arXiv:2608.21680 (Cowan, nonnesting permutations avoiding 123), arXiv:2608.21351 (Laudone, canon permutations), arXiv:2608.30002 (Shankar, canon permutations) and arXiv:2312.16052; none treats the Table 4 rows except that Demonstrandum Research published in July 2026 a Lean proof of the row {1132,3312}. This is a bounded negative finding for the other rows.