Congruences for Dedekind numbers

Abstract

We study congruence properties of Dedekind numbers $d_n$. Our approach relies on actions of $p$-subgroups of symmetric groups on posets of monotone functions and leads to the congruences $d_{10}\equiv 106(\mathrm{mod}210)$, $d_{11}\equiv 117(\mathrm{mod}210)$, and $d_{12}\equiv 12(\mathrm{mod}30)$. In particular, $d_{11}$ is odd, confirming a claim stated by Yamamoto without a proof.