Kurkov’s identity for ordered set partitions

For every n>0, prove or disprove that the Fubini number a(n), counting ordered set partitions, equals Σₖ₌₀^(2^(n−1)−1)A284005(k). The combinatorial approach sorts partitions by their prescribed block minima.

Why it matters and what counts as progress

Why it matters: It asks for a uniform explanation of an identity relating two enumerative sequences rather than a finite numerical check.

What counts as progress: A proof of the ordered-partition identity for every n, with a combinatorial interpretation of each summand.

How the problem developed

8 Jul 2018 · Conjecture [1]
Mikhail Kurkov Submitted to OEIS the conjectured formula expressing Fubini numbers as partial sums of sequence A284005.
July 2026 · Proof [1]
John Erlbacher Published an elementary proof of Kurkov’s identity for all positive n, with a refined count by block-minimum classes.

AI contribution

A proof of Kurkov's conjecture for A000670

John Erlbacher

For every n>0, the Fubini number a(n), which counts ordered set partitions, equals the sum of A284005(k) for k from 0 through 2^(n−1)−1. The proposed proof counts partitions with each prescribed set of block minima.

AI involvementAI-centered

ImportanceModest

Technical verificationPartial

Human understandingFew