bibkey: kurkov2025a126347 authors: Paul D. Hanna; Mikhail Kurkov year: 2025 title: “OEIS A126347 q-Bell coefficient triangle and Kurkov’s reversed-row conjecture” doi: null url: https://oeis.org/A126347 claim: “A126347 %N: Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n). A126347 %F: Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025” strata_touched:

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

OEIS A126347: q-Bell rows and Kurkov’s recurrence

This note is about the OEIS entry: its contributors are Paul D. Hanna (%A) and Mikhail Kurkov, who posted the %F conjecture. Carl G. Wagner and the pair Jianping Pan and Tianyi Yu are cited below as secondary sources that attest the recurrence and the degree fact; they are not contributors to this entry and not coauthors of it.

OEIS A126347 records the coefficient rows of the q-Bell polynomials and attributes the displayed recurrence to the literature through Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), Journal of Integer Sequences 7 (2004), Article 04.1.1. Jianping Pan and Tianyi Yu, Top-degree components of Grothendieck and Lascoux polynomials, arXiv:2302.03643v4 (journal version 2024), also state this recurrence and attribute it to Wagner.

Pan and Yu identify the reversed polynomial as q^(binom(n,2)) B_n(q^(-1)). Thus the triangular degree and monicity of B_n are literature-attested facts. The formal module proves them locally because the pinned Lean libraries do not provide this q-Bell family, but does not present them as the new intermediate content.

Verified locator

The following lines were read from the OEIS text endpoint on 2026-09-16.

• URL: https://oeis.org/A126347
• %O: 0,6
• %N (verbatim): Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).
• Conjecture %F (verbatim): Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025
• %A (verbatim): Paul D. Hanna, Dec 31 2006, May 28 2007

The Wagner paper predates Kurkov’s conjecture and does not contain the triangular recurrence R. Pan and Yu likewise contain no occurrence of Kurkov or “row polynomial” and predate the conjecture. They attest the q-Bell recurrence and its reversal degree, not the identity between that reversal and Kurkov’s diagonal.

Formal scope

The formal result concerns only Kurkov’s quoted recurrence and the next q-Bell row. Mathlib defines Polynomial.reverse f by reflection about f.natDegree; monicity and degree binom(n,2) make this equal to reversal of the fixed-length OEIS coefficient row. Without that fact, leading zero slots in a fixed row would not be represented by Polynomial.reverse.

The formal kurkovR is total: row zero is zero away from column zero, and later recurrence clauses accept every natural column. The diagonal at row n+1 reads only columns j<n+1 of row n, so values outside the source domain k<=n do not enter the theorem.