On the Bell monoid

The Plactic monoid is undoubtedly a very important algebraic structure, mainly because of its connections with a wide variety of topics in Mathematics, such as the theory of Schur functions, Kostka-Foulkes polynomials, Yang-Baxter equations and more recently Kashiwara’s theory of crystal bases. The Plactic monoid has also motivated the study of other combinatorial monoids, like the Sylvester, the hypoplactic, the Chinese, the Baxter and the Bell monoids. In this presentation we will be interested in the last one. The Bell monoid was originally presented via an insertion algorithm on words, which then gives rise to an equivalence by identifying words leading to the same tableau. In the same way we present a new insertion algorithm and a new congruence coinciding and that generate the Bell monoid. Motivated by recent developments regarding conjugacy in Sylvester monoids, we will present several results both in the Bell monoid and in its restriction to permutations. Formulas to count the number of their elements satisfying some conditions will also be presented.