107 research outputs found
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 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
Permutads
We unravel the algebraic structure which controls the various ways of
computing the word ((xy)(zt)) and its siblings. We show that it gives rise to a
new type of operads, that we call permutads. It turns out that this notion is
equivalent to the notion of "shuffle algebra" introduced by the second author.
It is also very close to the notion of "shuffle operad" introduced by V.
Dotsenko and A. Khoroshkin. It can be seen as a noncommutative version of the
notion of nonsymmetric operads. We show that the role of the associahedron in
the theory of operads is played by the permutohedron in the theory of
permutads.Comment: Same results, re-arranged and more details. 38 page
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