Pattern avoidance in labelled trees

We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words, permutations, coloured permutations etc. The notion of Wilf equivalence for patterns in permutations admits a straightforward generalisation for (sets of) tree patterns; we describe classes for trees with small numbers of leaves, and give several bijections between trees avoiding pattern sets from the same class. We also explain a few general results for tree pattern avoidance, both for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining the operadic contex

Combinatorial generation via permutation languages. VI. Binary trees

In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-M\"utze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions

Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.Comment: 14 page