25,980 research outputs found
Maximum size of reverse-free sets of permutations
Two words have a reverse if they have the same pair of distinct letters on
the same pair of positions, but in reversed order. A set of words no two of
which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size
of a reverse-free set of words from [n]^k where no letter repeats within a
word. We show the following lower and upper bounds in the case n >= k: F(n,k)
\in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of
n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.Comment: 10 page
Path separation by short cycles
Two Hamilton paths in are separated by a cycle of length if their
union contains such a cycle. For small fixed values of we bound the
asymptotics of the maximum cardinality of a family of Hamilton paths in
such that any pair of paths in the family is separated by a cycle of length
Comment: final version with correction
Block Crossings in Storyline Visualizations
Storyline visualizations help visualize encounters of the characters in a
story over time. Each character is represented by an x-monotone curve that goes
from left to right. A meeting is represented by having the characters that
participate in the meeting run close together for some time. In order to keep
the visual complexity low, rather than just minimizing pairwise crossings of
curves, we propose to count block crossings, that is, pairs of intersecting
bundles of lines.
Our main results are as follows. We show that minimizing the number of block
crossings is NP-hard, and we develop, for meetings of bounded size, a
constant-factor approximation. We also present two fixed-parameter algorithms
and, for meetings of size 2, a greedy heuristic that we evaluate
experimentally.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
A bijection to count (1-23-4)-avoiding permutations
A permutation is (1-23-4)-avoiding if it contains no four entries, increasing
left to right, with the middle two adjacent in the permutation. Here we give a
2-variable recurrence for the number of such permutations, improving on the
previously known 4-variable recurrence. At the heart of the proof is a
bijection from (1-23-4)-avoiding permutations to increasing ordered trees whose
leaves, taken in preorder, are also increasing.Comment: latex, 16 page
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
Cryptanalysis of SIGABA
SIGABA is a World War II cipher machine used by the United States. Both the United States Army and the United States Navy used it for tactical communication. In this paper, we consider an attack on SIGABA using the largest practical keyspace for the machine. This attack will highlight the strengths and weaknesses of the machine, as well as provide an insight into the strength of the security provided by the cipher
Geometrically constructed bases for homology of partition lattices of types A, B and D
We use the theory of hyperplane arrangements to construct natural bases for
the homology of partition lattices of types A, B and D. This extends and
explains the "splitting basis" for the homology of the partition lattice given
in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the
following general technique is presented and utilized. Let A be a central and
essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions
of a generic hyperplane section of A. We show that there are induced polytopal
cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the
intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde
H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial
homology bases is applied to the Coxeter arrangements of types A, B and D, and
to some interpolating arrangements.Comment: 29 pages, 4 figure
Shifted symmetric functions and multirectangular coordinates of Young diagrams
In this paper, we study shifted Schur functions , as well as a
new family of shifted symmetric functions linked to Kostka
numbers. We prove that both are polynomials in multi-rectangular coordinates,
with nonnegative coefficients when written in terms of falling factorials. We
then propose a conjectural generalization to the Jack setting. This conjecture
is a lifting of Knop and Sahi's positivity result for usual Jack polynomials
and resembles recent conjectures of Lassalle. We prove our conjecture for
one-part partitions.Comment: 2nd version: minor modifications after referee comment
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