53,138 research outputs found
Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
The aim of this paper is to bring together the three objects in the title.
Recall that, given a Lie algebra , the Eulerian idempotent is a
canonical projection from the enveloping algebra to
. The Baker-Campbell-Hausdorff product and the Magnus expansion
can both be expressed in terms of the Eulerian idempotent, which makes it
interesting to establish explicit formulas for the latter. We show how to
reduce the computation of the Eulerian idempotent to the computation of a
logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The
problem of finding formulas for the pre-Lie logarithm, which is interesting in
its own right -- being related to operad theory, numerical analysis and
renormalization -- is addressed using techniques inspired by umbral calculus.
As a consequence of our analysis, we find formulas both for the Eulerian
idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.Comment: Preliminary version. Comments are welcome
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
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size in the binary search
tree and the random recursive tree (of total size ) asymptotically has a
Poisson distribution if , and that the distribution is
asymptotically normal for . Furthermore, we prove similar
results for the number of subtrees of size with some required property , for example the number of copies of a certain fixed subtree . Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of -protected nodes in a binary search tree or random recursive tree.
The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings
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