Associativity conditions for linear quasigroups and equivalence relations on binary trees

We characterise the bracketing identities satisfied by linear quasigroups with the help of certain equivalence relations on binary trees that are based on the left and right depths of the leaves modulo some integers. The numbers of equivalence classes of $n$-leaf binary trees are variants of the Catalan numbers, and they form the associative spectrum (a kind of measure of non-associativity) of a quasigroup.Comment: 32 page

Coloured peak algebras and Hopf algebras

For $G$ a finite abelian group, we study the properties of general equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of $G$ with the symmetric group \SG_n, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of \k G_n as well as graded connected Hopf subalgebras of \bigoplus_{n\ge o} \k G_n. In particular we construct a $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Hopf algebra. We also have similar results for a $G$-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the $G$-coloured descent Hopf algebra whose image is the $G$-coloured peak Hopf algebra. We outline a theory of combinatorial $G$-coloured Hopf algebra for which the $G$-coloured quasi-symmetric Hopf algebra and the graded dual to the $G$-coloured peak Hopf algebra are central objects.Comment: 26 pages latex2

On super-strong Wilf equivalence classes of permutations

Super-strong Wilf equivalence is a type of Wilf equivalence on words that was originally introduced as strong Wilf equivalence by Kitaev et al. [Electron. J. Combin. 16(2)] in 2009. We provide a necessary and sufficient condition for two permutations in n letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in the same article by Kitaev et al. that the cardinality of each Wilf equivalence class is a power of 2

Pattern avoidance in binary trees

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns.Comment: 19 pages, many images; published versio