The YY game

We introduce a new one-person game similar to the Sudoku game. It is based on combinatorial objects called planar binary rooted trees. It is related to the four color conjecture. Its mathematical analysis makes use of the Tamari poset, hence the Stasheff associahedron.Comment: 7 page

Quadri-algebras

We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an infinitesimal bialgebra A. This algebra carries a canonical pair of commuting Baxter operators: \beta(T)=T\ast\id and \gamma(T)=\id\ast T, where $\ast$ denotes the convolution of endomorphisms. It follows that End(A) is a quadri-algebra, whenever A is an infinitesimal bialgebra. We also discuss commutative quadri-algebras and state some conjectures on the free quadri-algebra