### A bijection between atomic partitions and unsplitable partitions.

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The purpose of this article is to give, for any (commutative) ring $A$, an explicit minimal set of generators for the ring of multisymmetric functions ${\mathrm{T}S}_{A}^{d}\left(A[{x}_{1},\cdots ,{x}_{r}]\right)={\left(A{[{x}_{1},\cdots ,{x}_{r}]}^{{\otimes}_{A}d}\right)}^{{\U0001d516}_{d}}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a $\mathbb{Q}$-algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously...

A direct formula for jeu de taquin applied to the swap of two rows of standard tableaux is given. A generalization of this formula to non standard tableaux is used to describe combinatorially a path basis isomorphism for the algebra of type ${A}_{l}$.