The simplest Young diagrams and groups generated by their tableaux

To study the combinatorics of Young diagrams, Anatoly Moiseevich Vershik proposed to consider groups generated by natural involutions of two-row Young tables. The definitions and the first results were described by A. M. Vershik and N. V. Tsilevich. In this paper the class of the studied diagrams is expanded and new methods for their characterization are given.