A double bialgebra is a family $(A,m,\Delta,\delta)$ such that both $(A,m,\Delta)$ and $(A,m,\delta)$ are bialgebras, with the extra condition that seeing $\delta$ as a right coaction on itself,
$m$ and $\Delta$ are right comodules morphism over $(A,m,\delta)$. A classical example is given by the polynomial algebra $\mathbb{C}[X]$, with its two classical coproducts.
In this talk, we will present a double bialgebra structure on the symmetric algebra generated by noncrossing partitions. The first coproduct is given by the separations of the blocks of the partitions,
with respect to the entanglement, and the second one by fusions of blocks.
This structure implies that there exists a unique polynomial invariant on noncrossing partitions which respects both coproducts: we will give some elements on this invariant,
and applications to the antipode of noncrossing partitions.