113 research outputs found
Non overlapping partitions, continued fractions, Bessel functions and a divergent series
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Generalised Pattern Avoidance
Recently, Babson and Steingrimsson have introduced generalised permutation
patterns that allow the requirement that two adjacent letters in a pattern must
be adjacent in the permutation. We consider pattern avoidance for such
patterns, and give a complete solution for the number of permutations avoiding
any single pattern of length three with exactly one adjacent pair of letters.
We also give some results for the number of permutations avoiding two different
patterns. Relations are exhibited to several well studied combinatorial
structures, such as set partitions, Dyck paths, Motzkin paths, and involutions.
Furthermore, a new class of set partitions, called monotone partitions, is
defined and shown to be in one-to-one correspondence with non-overlapping
partitions
Generating functions for generating trees
Certain families of combinatorial objects admit recursive descriptions in
terms of generating trees: each node of the tree corresponds to an object, and
the branch leading to the node encodes the choices made in the construction of
the object. Generating trees lead to a fast computation of enumeration
sequences (sometimes, to explicit formulae as well) and provide efficient
random generation algorithms. We investigate the links between the structural
properties of the rewriting rules defining such trees and the rationality,
algebraicity, or transcendence of the corresponding generating function.Comment: This article corresponds, up to minor typo corrections, to the
article submitted to Discrete Mathematics (Elsevier) in Nov. 1999, and
published in its vol. 246(1-3), March 2002, pp. 29-5
ZEONS, LATTICES OF PARTITIONS, AND FREE PROBABILITY
International audienceCentral to the theory of free probability is the notion of summing multiplicative functionals on the lattice of non-crossing partitions. In this paper, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of “zeon” algebras (commutative subalgebras of fermion algebras), multiplicative functions can be summed over segments of lattices of partitions by employing methods of “zeon-Berezin” operator calculus. In particular, properties of the algebra are used to “sieve out” the appropriate segments and sub-lattices. The work concludes with an application to joint moments of quantum random variables
Continued fraction expansions for q-tangent and q-cotangent functions
For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical
Constants (2003) and Mathematical Constants II (2019), both published by
Cambridge University Press. Comments are always welcome.Comment: 162 page
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