slug: oeis-a126347-kurkov-reversed-qbell-row-polynomial bibkey: kurkov2025a126347 doi: null url: https://oeis.org/A126347 triage: theorem motivation_gids:

• D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.result

Kurkov’s reversed q-Bell row polynomial

Problem

OEIS A126347 has offset %O 0,6. Its coefficient data begins (%S, verbatim):

1,1,1,1,1,2,1,1,1,3,3,4,2,1,1,1,4,6,10,9,7,7,4,2,1,1,1,5,10,20,25,26,

Its NAME line (%N, verbatim) is:

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).

Mikhail Kurkov’s conjecture (%F, verbatim) is:

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

The AUTHOR line (%A, verbatim) is:

Paul D. Hanna, Dec 31 2006, May 28 2007

The referenced Bell-number entry A000110 has NAME line (%N, verbatim):

Bell or exponential numbers: number of ways to partition a set of n labeled elements.

The exact formal statement is:

forall n : Nat, kurkovR n n = (qBell (n + 1)).reverse

Here qBell uses Wagner’s q-Bell recurrence. The formal reversal is reflection about the natural degree. The locally proved facts that qBell n is monic of degree Nat.choose n 2 ensure that this is the fixed-row coefficient reversal meant by the source.

Motivation

Kurkov’s recurrence builds each row from the preceding diagonal and its prefix sums. The theorem identifies its diagonal with the reversed q-Bell coefficient rows for every natural index, upgrading the printed conjecture from numerical evidence to a kernel-checked unbounded identity.

Gap

The OEIS text endpoint still labels the line “Conjecture”. Wagner’s 2004 paper establishes the q-Bell recurrence but predates Kurkov’s triangular recurrence. Pan and Yu, arXiv:2302.03643v4, identify the degree-based reversal of q-Bell polynomials and thereby attest their degree and monicity, but their text contains neither Kurkov’s recurrence nor the conjectured identity. No dominating theorem was found in the repository or pinned Mathlib searches recorded in issue #8219.

The load-bearing new proposition is kurkovR_pascal_expansion: it expands a Kurkov row prefix into a binomial Pascal row. This proposition is neither a restatement of the final diagonal identity nor a direct instance of the available polynomial library, and it remains on the final proof path.

The result formalizes only Kurkov’s quoted conjecture and no other formula or comment line from the A126347 entry.

Route

1. Prove by strong induction that qBell n is monic of triangular degree.
2. Prove the Pascal expansion of kurkovR n k by induction on the row and column, using Pascal’s identity for the binomial weights.
3. Reflect Wagner’s recurrence about the triangular degree and identify the reflected convolution with the Pascal expansion.
4. Induct on the diagonal and conclude with Mathlib’s degree-based reverse.

Falsifier

Any natural n for which kurkovR n n differs from the coefficient reversal of qBell (n+1) contradicts the theorem. A failure of monicity or triangular degree would also invalidate the identification of Mathlib’s natural-degree reverse with the source’s fixed-row reversal.

Evidence

• qBell_monic_degree proves monicity and degree Nat.choose n 2 for every row.
• kurkovR_pascal_expansion proves the binomial expansion of every in-range Kurkov row entry and is used in the diagonal induction.
• result has exactly the universal statement displayed in the Problem section.
• Wagner 2004 and Pan–Yu 2024 attest the q-Bell recurrence and reversal degree, while neither contains Kurkov’s later triangular identity.

Triage

theorem; resolution proved.

ASSUMED-UNVERIFIED

Historical priority outside the named OEIS, Wagner, Pan–Yu, repository, and pinned-library surfaces is unverified. No exhaustive novelty or priority claim is made; this boundary does not affect the formal universal identity.