164 research outputs found
Compatible associative products and trees
We compute dimensions of graded components for free algebras with two
compatible associative products, and give a combinatorial interpretation of
these algebras in terms of planar rooted trees.Comment: 19 pages, added a note on relation to other operad
Freeness theorems for operads via Gr\"obner bases
We show how to use Groebner bases for operads to prove various freeness
theorems: freeness of certain operads as nonsymmetric operads, freeness of an
operad Q as a P-module for an inclusion P into Q, freeness of a suboperad. This
gives new proofs of many known results of this type and helps to prove some new
results.Comment: 15 pages, no figures, corrected typos and changed in parts the
structure of the pape
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
The third parafermionic chiral algebra with the symmetry Z_{3}
We have constructed the parafermionic chiral algebra with the principal
parafermionic fields \Psi,\Psi^{+} having the conformal dimension
\Delta_{\Psi}=8/3 and realizing the symmetry Z_{3}.Comment: 6 pages, no figur
Spin--spin critical point correlation functions for the 2D random bond Ising and Potts models
We compute the combined two and three loop order correction to the spin-spin
correlation functions for the 2D Ising and q-states Potts model with random
bonds at the critical point. The procedure employed is the renormalisation
group approach for the perturbation series around the conformal field theories
representing the pure models. We obtain corrections for the correlations
functions which produce crossover in the amplitude but don't change the
critical exponent in the case of the Ising model and which produce a shift in
the critical exponent, due to randomness, in the case of the Potts model.
Comparison with numerical data is discussed briefly.Comment: 10pp, latex, PAR--LPTHE 94/1
Anick-type resolutions and consecutive pattern avoidance
For permutations avoiding consecutive patterns from a given set, we present a
combinatorial formula for the multiplicative inverse of the corresponding
exponential generating function. The formula comes from homological algebra
considerations in the same sense as the corresponding inversion formula for
avoiding word patterns comes from the well known Anick's resolution.Comment: 16 pages. Preliminary version, comments are welcom
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