Kourovka Problem 18.50 — prescribed permuted-product cardinality

S. Kohl posed the problem on MathOverflow in March 2013; it became Problem $18.50$ in the eighteenth Kourovka Notebook, published in 2014. The question asks whether, for every $n$ and every $k$ from $1$ through $n!$, one can choose $n$ distinct group elements whose permuted products have exactly $k$ values.

Known results

• The assertion was computationally known for $n\leq 4$.
• Benjamin Young, 2013, gave examples in $S_5$.
• Kohl’s discussion recorded the remaining need to prove that no unintended product equalities occurred.

2026 affirmative solution

An arXiv paper proves the theorem for all $n$ and $1\leq k\leq n!$. It constructs a central extension of $\mathbb{Z}^n$ by $\mathbb{Z}/k\mathbb{Z}$ in which the permuted products encode permutations by inversion number modulo $k$, producing exactly $k$ values. The paper states that Aristotle, developed by Harmonic, autonomously discovered and formally verified the solution in Lean.

Current status (as of July 2026): The problem is settled affirmatively for every positive integer $n$ and every $k$ with $1\leq k\leq n!$.