bibkey: kurkov2024a003319 authors: Mikhail Kurkov year: 2024 title: “OEIS A003319 and Kurkov’s A370380/A370381 double-array conjecture” doi: null url: https://oeis.org/A003319 claim: “Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - Mikhail Kurkov, Apr 26 2024” strata_touched:

• D5/S1/Recurrence/Algebraic/IndecomposablePermutationDoubleArray license: citation-only triage: anchor

OEIS A003319 and Kurkov’s two arrays

Verified locator

The source check supplied for this formalization read these immutable OEIS-data revisions:

• Canonical entry: https://oeis.org/A003319
• A003319 at https://github.com/oeis/oeisdata/blob/dacce77176c40534c4309db03464e7f4c08d83dd/seq/A003/A003319.seq
• A370380 at https://github.com/oeis/oeisdata/blob/a32b213583dbdd91a31cc0975e203471b3afe0fb/seq/A370/A370380.seq
• A370381 at https://github.com/oeis/oeisdata/blob/ed2582877dea2190aa163abe737afe7d42179fae/seq/A370/A370381.seq

The A003319 conjecture line is, verbatim:

Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1.

It is attributed there to Mikhail Kurkov on April 26, 2024. The sequence counts permutations of {1,...,n} which preserve no proper nonempty initial interval. The formal subtype uses Fin n, so its interval {0,...,k-1} is the same condition after translating labels by one. The formal c(n) is the actual finite cardinality of this subtype.

The two referenced entries define arrays with natural row and column indices:

U(0,k) = 1
U(m+1,k) = (k+2) U(m,k+1) + Sum_{j=0..k} U(m,j)

V(0,k) = 1
V(m+1,k) = Sum_{j=0..k+1} binomial(k+2,j+1) V(m,j)

These are formalized directly as U and V; neither array is used to define the permutation count.

Scalar recurrence sources

The supplied source audit read the complete 19-page author manuscript https://plouffe.fr/OEIS/citations/indec.pdf, SHA-256 ff0934dc8f7a404cfa2b01b698fbc9d5ced75a9087e14edb64df4be8f92ffb25. Its equation (2) is the first-block recurrence and its Theorem 1 gives the equivalent positive scalar recurrence for indecomposable permutations. The inspected manuscript contains no identification with either Kurkov array.

The supplied audit also read printed pages 10–12 of https://arxiv.org/pdf/0912.0573, SHA-256 f8c17f417b1b5239d8bce4ee5faf3fb02ced6137dbda5746fdb31d995af38ee6. Hetyei attributes c(n)=Sum_{i=1..n-1} i*c(i)*(n-i-1)! to King and reconstructs the same permutation decomposition. Substitution i=n-j-1 identifies the two scalar forms. These scalar facts are known mathematics and are proved only locally as the bridge from the actual cardinality to the array argument.

Source status and scope

The supplied audit read all 27 recorded A370380 revisions, all 15 A370381 revisions, A003319 revisions 413–462, the Sondow proof in the entry, and Barry arXiv:1804.06801v1. It found no proof of either array identity in those materials. Current conjecture labels and bounded search misses do not establish novelty or priority.

The complete final Elsevier body corresponding to King’s manuscript was not directly inspected, and no manuscript/final equivalence or worldwide-priority claim is made. The author manuscript and Hetyei establish the scalar first-block recurrence, not the two array-border identities settled by the formal result.