Kourovka Notebook Problem 20.25

Definitions and mathematical statement

For each positive integer $k$, let $n=k^2$, and let $S_n$ be the symmetric group on $n$ letters. Choose $\sigma_k\in S_{k^2}$ whose disjoint cycle lengths are exactly $1,3,5,\ldots,2k-1$, each occurring once; these lengths sum to $k^2$. Interpret the character table as the ordinary complex irreducible character table. Its rows are indexed by partitions $\lambda\vdash k^2$; write $\chi^\lambda$ for the corresponding character and $p(m)$ for the number of partitions of $m$. The values $\chi^\lambda(\sigma_k)$ are integers. An entry is odd if it belongs to $2{\mathbb{Z}}+1$, including negative odd integers. Define \[ a_k=\frac{\#\{\lambda\vdash k^2:\chi^\lambda(\sigma_k)\in 2{\mathbb{Z}}+1\}}{p(k^2)}. \] This proportion is independent of the choice of representative $\sigma_k$ of the specified conjugacy class. Determine \[ \lim_{k\to\infty}a_k, \] including establishing its existence. The source asks for a limiting proportion without separately asserting that the limit exists; the ordinary complex character-table interpretation is made explicit here.

Short English statement

For permutations in $S_{k^2}$ with one cycle of each odd length from $1$ to $2k-1$, determine the limiting fraction of irreducible character values that are odd integers as $k\to\infty$. This is the problem attributed to D. Gluck; the supplied registry labels it open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701683$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 20.25. Its supplied status is open; this label is not an independently verified literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem number 20.25, proposed by D. Gluck in 2022. The supplied provenance identifies the 20th issue (2022) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Background reference supplied in the statement: D. Gluck, Proc. Amer. Math. Soc. 147 (2019), 1005--1011, specifically pages 1005--1006. The article title and a direct link are not supplied.