Zero-free columns in character tables of symmetric groups
Colin Defant, Sidharth Hariharan, Kenny Lau and Ken Ono
Preprint
August 2026
DOI: 10.48550/arXiv.2608.27718
The rows and columns of the character table of the symmetric group \(S_n\) are both naturally indexed by partitions of \(n\). Let \(D(n)\) denote the number of conjugacy classes of \(S_n\) whose column contains no zero entry. The identity column is always zero-free, so \(D(n) \geq 1\). It is known that \(D(n) \ll n^2\). We prove that \(D(n) \ll n^{3/4}\). Second, we prove for almost all positive integers \(n\) that \(D(n) \ll_B n^{1/2}(\log n)^B\) for every \(B > 5/6\), with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that \(D(n) \ll_{\varepsilon} n^{\varepsilon}\). AxiomProver formalized the results in this paper in Lean assuming preexisting literature.