  Mixed radix bases in numeration is a very old notion but it is rarely studied
on its own or in relation with concrete problems related to number theory.
Starting from the natural question of the conversion of a basis to another for
integers as well as polynomials, we use mixed radix bases to introduce
two-dimensional arrays with suitable filling rules. These arrays provide
algorithms of conversion which uses only a finite number of euclidean division
to convert from one basis to another; it is interesting to note that these
algorithms are generalizations of the well-known H\"orner's rule of quick
evaluation of polynomials. The two-dimensional arrays with local
transformations are reminiscent from statistical mechanics models: we show that
changes between three numeration basis are related to the set-theoretical
Yang-Baxter equation and this is, up to our knowledge, the first time that such
a structure is described in number theory. As an illustration, we reinterpret
well-known results around Furstenberg's conjecture in terms of Yang-Baxter
transformations between mixed radix bases, hence opening the way to alternative
approaches.
