22,186 research outputs found
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
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
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
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