18 research outputs found
Multidimensional cellular automata and generalization of Fekete's lemma
Fekete's lemma is a well known combinatorial result on number sequences: we
extend it to functions defined on -tuples of integers. As an application of
the new variant, we show that nonsurjective -dimensional cellular automata
are characterized by loss of arbitrarily much information on finite supports,
at a growth rate greater than that of the support's boundary determined by the
automaton's neighbourhood index.Comment: 6 pages, no figures, LaTeX. Improved some explanations; revised
structure; added examples; renamed "hypercubes" into "right polytopes"; added
references to Arratia's paper on EJC, Calude's book, Cook's proof of Rule 110
universality, and arXiv paper 0709.117
Shape Avoiding Permutations
Permutations avoiding all patterns of a given shape (in the sense of
Robinson-Schensted-Knuth) are considered. We show that the shapes of all such
permutations are contained in a suitable thick hook, and deduce an exponential
growth rate for their number.Comment: 16 pages; final form, to appear in J. Combin. Theory, Series
Counting occurrences of 132 in an even permutation
We study the generating function for the number of even (or odd) permutations
on n letters containing exactly r\gs0 occurrences of 132. It is shown that
finding this function for a given r amounts to a routine check of all
permutations in .Comment: 12 pages, 2 figure
Pattern Avoidance in Poset Permutations
We extend the concept of pattern avoidance in permutations on a totally
ordered set to pattern avoidance in permutations on partially ordered sets. The
number of permutations on that avoid the pattern is denoted
. We extend a proof of Simion and Schmidt to show that for any poset , and we exactly classify the posets for which
equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and
improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated
author email addresse
Asymptotic enumeration of permutations avoiding generalized patterns
Motivated by the recent proof of the Stanley-Wilf conjecture, we study the
asymptotic behavior of the number of permutations avoiding a generalized
pattern. Generalized patterns allow the requirement that some pairs of letters
must be adjacent in an occurrence of the pattern in the permutation, and
consecutive patterns are a particular case of them.
We determine the asymptotic behavior of the number of permutations avoiding a
consecutive pattern, showing that they are an exponentially small proportion of
the total number of permutations. For some other generalized patterns we give
partial results, showing that the number of permutations avoiding them grows
faster than for classical patterns but more slowly than for consecutive
patterns.Comment: 14 pages, 3 figures, to be published in Adv. in Appl. Mat
Improved bounds and new techniques for Davenport-Schinzel sequences and their generalizations
Let lambda_s(n) denote the maximum length of a Davenport-Schinzel sequence of
order s on n symbols. For s=3 it is known that lambda_3(n) = Theta(n alpha(n))
(Hart and Sharir, 1986). For general s>=4 there are almost-tight upper and
lower bounds, both of the form n * 2^poly(alpha(n)) (Agarwal, Sharir, and Shor,
1989). Our first result is an improvement of the upper-bound technique of
Agarwal et al. We obtain improved upper bounds for s>=6, which are tight for
even s up to lower-order terms in the exponent. More importantly, we also
present a new technique for deriving upper bounds for lambda_s(n). With this
new technique we: (1) re-derive the upper bound of lambda_3(n) <= 2n alpha(n) +
O(n sqrt alpha(n)) (first shown by Klazar, 1999); (2) re-derive our own new
upper bounds for general s; and (3) obtain improved upper bounds for the
generalized Davenport-Schinzel sequences considered by Adamec, Klazar, and
Valtr (1992). Regarding lower bounds, we show that lambda_3(n) >= 2n alpha(n) -
O(n), and therefore, the coefficient 2 is tight. We also present a simpler
version of the construction of Agarwal, Sharir, and Shor that achieves the
known lower bounds for even s>=4.Comment: To appear in Journal of the ACM. 48 pages, 3 figure
Degrees of nonlinearity in forbidden 0–1 matrix problems
AbstractA 0–1 matrix A is said to avoid a forbidden 0–1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport–Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0–1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n×n matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit n×n matrices with weight Θ(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0–1 matrices. In the second part of the article we simplify one aspect of Keszegh and Geneson’s proof that there are infinitely many minimal nonlinear forbidden 0–1 matrices. In the last part of the article we investigate the relationship between 0–1 matrices and generalized Davenport–Schinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function