Forbidden Subsequences Research Articles

THE ANALYSIS OF RANDOM PERMUTATIONS WITH s-PROHIBITION

Transpositions of lower lines of random permutanions of size n with forbidden subsequences of size s < n are considered. The number of such transpositions are finded, their enumerations are realized, the numeration problem of outcomes of the scheme is decided and their modelling is considered.

Forbidden traces and forbidden subtraces

The concepts of forbidden strings and forbidden subsequences are generalized to traces. This paper presents algorithms for constructing sets of minimal forbidden traces and minimal forbidden subtraces for a given trace.

On the modular sumset partition problem

A sequence m1≥m2≥⋯≥mk of k positive integers isn-realizable if there is a partition X1,X2,…,Xk of the integer interval [1,n] such that the sum of the elements in Xi is mi for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)3, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.

Degrees of nonlinearity in forbidden 0–1 matrix problems

A 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.

Origins of Nonlinearity in Davenport–Schinzel Sequences

A generalized Davenport-Schinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let $\mathrm{Ex}(\sigma,n)$ be the maximum length of a sequence over an $n$-letter alphabet excluding subsequences isomorphic to $\sigma$. It has been proved that for every $\sigma$, $\mathrm{Ex}(\sigma,n)$ is either linear or very close to linear. In particular it is $O(n2^{\alpha(n)^{O(1)}})$, where $\alpha$ is the inverse-Ackermann function and $O(1)$ depends on $\sigma$. In much the same way that the complete graphs $K_5$ and $K_{3,3}$ represent the minimal causes of nonplanarity, there must exist a set $\Phi_{Nonlin}$ of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of $\Phi_{Nonlin}$. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that $\Phi_{Nonlin}$ is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences.

Construction of Maximum Run-Length Limited Codes Using Sequence Replacement Techniques

The sequence replacement technique converts an input sequence into a constrained sequence in which a prescribed subsequence is forbidden to occur. Several coding algorithms are presented that use this technique for the construction of maximum run-length limited sequences. The proposed algorithms show how all forbidden subsequences can be successively or iteratively removed to obtain a constrained sequence and how special subsequences can be inserted at predefined positions in the constrained sequence to represent the indices of the positions where the forbidden subsequences were removed. Several modifications are presented to reduce the impact of transmission errors on the decoding operation, and schemes to provide error control are discussed as well. The proposed algorithms can be implemented efficiently, and the rates of the constructed codes are close to their theoretical maximum. As such, the proposed algorithms are of interest for storage systems and data networks.

A note on certain de Bruijn sequences with forbidden subsequences

We consider words over a finite alphabet with certain uniqueness properties (a subsequence of length k does not occur more than once) and distance properties (at least j other symbols separate the occurrence of the same symbol). The maximal length of these words is realised by linear de Bruijn sequences with certain forbidden subsequences. We prove the existence of these maximal sequences.

Wilf-equivalence for singleton classes

Write p 1 p 2 … p m for the permutation matrix ( δ p i , j ) m × m . Let S n ( M ) be the set of n × n permutation matrices which do not contain the m × m permutation matrix M as a submatrix. In [R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383–406] Simion and Schmidt show bijectively that | S n ( 123 ) | = | S n ( 213 ) | . In the present work, we give a bijection from S n ( 12 … t p t + 1 … p m ) to S n ( t … 21 p t + 1 … p m ) . This result was established for t = 2 in [J. West, Permutations with forbidden subsequences and stack-sortable permutations, PhD thesis, MIT, Cambridge, MA, 1990] and for t = 3 in [E. Babson, J. West, The permutations 123 p 4 … p t and 321 p 4 … p t are Wilf-equivalent, Graphs Combin. 16 (2001) 373–380]. Moreover, if we think of n × n permutation matrices as transversals of the n by n square diagram, then we generalise this result to transversals of Young diagrams.

A bijection between permutations and floorplans, and its applications

A floorplan represents the relative relations between modules on an integrated circuit. Floorplans are commonly classified as slicing, mosaic, or general. Separable and Baxter permutations are classes of permutations that can be defined in terms of forbidden subsequences. It is known that the number of slicing floorplans equals the number of separable permutations and that the number of mosaic floorplans equals the number of Baxter permutations [B. Yao, H. Chen, C.K. Cheng, R.L. Graham, Floorplan representations: complexity and connections, ACM Trans. Design Automation Electron. Systems 8(1) (2003) 55–80]. We present a simple and efficient bijection between Baxter permutations and mosaic floorplans with applications to integrated circuits design. Moreover, this bijection has two additional merits: (1) It also maps between separable permutations and slicing floorplans; and (2) it suggests enumerations of mosaic floorplans according to various structural parameters.

Postscript: “Permutations with forbidden subsequences and a generalized Schröder number” [Discrete Mathematics 218 (2000) 121–130

Postscript: “Permutations with forbidden subsequences and a generalized Schröder number” [Discrete Mathematics 218 (2000) 121–130

Posets of Matrices and Permutations with Forbidden Subsequences

The enumeration of permutations with specific forbidden subsequences has applications in areas ranging from algebraic geometry to the study of sorting algorithms. We consider a ranked poset of permutation matrices whose global structure incorporates the solution to the equivalent problem of enumerating permutations which contain a required subsequence. We describe this structure completely for saturated chains of lengths one and two, so settling several new and general instances of the original problem, and conclude with a superficial asymptotic investigation of arbitrary chains whose length is small by comparison with the rank of its constituent permutations. The value of this approach is reflected in the appearance of closed polynomial formulae (related to the Robinson-Schensted correspondence) and of a framework for the systematic analysis of associated combinatorial questions; indeed, we begin by studying a simpler poset of 0-1 sequences as the natural environment in which to introduce our insertion and deletion operators.

Software Tools for DNA Sequence Design

The design of DNA sequences is a key problem for implementing molecular self-assembly with nucleic acid molecules. These molecules must meet several physical, chemical and logical requirements, mainly to avoid mishybridization. Since manual selection of proper sequences is too time-consuming for more than a handful of molecules, the aid of computer programs is advisable. In this paper two software tools for designing DNA sequences are presented, the DNASequenceGenerator and the DNASequenceCompiler. Both employ an approach of sequence dissimilarity based on the uniqueness of overlapping subsequences and a graph based algorithm for sequence generation. Other sequence properties like melting temperature or forbidden subsequences are also regarded, but not secondary structure errors or equilibrium chemistry. Fields of application are DNA computing and DNA-based nanotechnology. In the second part of this paper, sequences generated with the DNASequenceGenerator are compared to those from several publications of other groups, an example application for the DNASequenceCompiler is presented, and the advantages and disadvantages of the presented approach are discussed.

Stirling numbers interpolation using permutations with forbidden subsequences

We present a family of number sequences which interpolates between the sequences B n, of Bell numbers, and n!. It is defined in terms of permutations with forbidden patterns or subsequences. The introduction, as a parameter, of the number m of right-to-left minima yields an interpolation between Stirling numbers of the second kind S(n,m) and of the first kind (signless) c(n,m). Moreover, q-counting the restricted permutations by special inversions gives an interpolation between variants of the usual q-analogues of these numbers.

Optimal Short-Term Scheduling of Multiproduct Single-Stage Batch Plants with Parallel Lines

The short-term scheduling of multiproduct single-stage batch plants with parallel lines involves the constraints concerning the release times and due dates of orders, as well as the sequence-dependent setup times and forbidden subsequences of production orders and the ready times of units. Taking the above constraints into consideration, this paper develops a short-term scheduling model for the multiproduct single-stage batch plants with parallel lines using the continuous-time domain representation and the notation of time slot. When the model is developed, the allocation of orders and units to time slots is represented by two sets of binary variables. The model is formulated as a mixed-integer linear programming (MILP) problem. It not only involves fewer binary variables than any other model based on the notation of time slot (Pinto, J. M.; Grossman, I. E. Ann. Oper. Res. 1998, 81, 433) but also can be used to optimize several types of objective functions. Based on the characteristics of the model and multiproduct single-stage batch plants, some heuristic rules are introduced. The rational employment of these heuristic rules can cut down the size of the model and has no effect on the optimality of the scheduling problem. A number of computational examples show that the optimal scheduling can be obtained by solving the MILP model in reasonable CPU time, especially when the scheduling problem involving several identical orders is considered.

Some permutations with forbidden subsequences and their inversion number

A permutation π avoids the subpattern τ iff π has no subsequence having all the same pairwise comparisons as τ, and we write π∈ S( τ). We examine the classes of permutations, S(321), S(321,3 1 ̄ 42) and S(4231,4132), enumerated, respectively by the famous Catalan, Motzkin and Schröder number sequences. We determine their generating functions according to their length, number of active sites and inversion number. We also find the average inversion number for each class. Finally, we describe some bijections between these classes of permutations and some classes of parallelogram polyominoes, from which we deduce some relations between the parameters of Motzkin and Schröder permutations.

Permutations with forbidden subsequences and a generalized Schröder number

Using the technique of generating trees, we prove that there are exactly 10 classes of pattern avoiding permutations enumerated by the large Schröder numbers. For each integer, m⩾1, a sequence which generalizes the Schröder and Catalan numbers is shown to enumerate m+2 2 classes of pattern avoiding permutations. Combinatorial interpretations in terms of binary trees and polyominoes and a generating function for these sequences are given.

From Motzkin to Catalan permutations

For every integer j⩾1, we define a class of permutations in terms of certain forbidden subsequences. For j=1, the corresponding permutations are counted by the Motzkin numbers, and for j=∞ (defined in the text), they are counted by the Catalan numbers. Each value of j>1 gives rise to a counting sequence that lies between the Motzkin and the Catalan numbers. We compute the generating function associated to these permutations according to several parameters. For every j⩾1, we show that only this generating function is algebraic according to the length of the permutations.

Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

Stack words stem from studies on stack-sortable permutations and represent classical combinatorial objects such as standard Young tableaux, permutations with forbidden sequences and planar maps. We extend existing enumerative results on stack words and we also obtain new results. In particular, we make a correspondence between nonseparable 3× n rectangular standard Young tableaux (or stack words where elements satisfy a ‘Towers of Hanoi’ condition) and nonseparable cubic rooted planar maps with 2 n vertices enumerated by 2 n (3 n)!/((2 n+1)!( n+1)!). Moreover, these tableaux without two consecutive integers in the same row are in bijection with nonseparable rooted planar maps with n+1 edges enumerated by 2(3 n)!/((2 n+1)!( n+1)!).

Permutations avoiding an increasing number of length-increasing forbidden subsequences

A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.

Directed animals, forests and permutations

In this paper, we illustrate a method to enumerate two-dimensional directed animals, with compact sources, on both the square and the triangular lattice. We give a recursive description of these structures from which we deduce their generating function, according to various parameters: the area, the right half-width and the number of compact sources. We determine the average number of compact sources in both lattices. Referring to single-source animals on the square lattice, we show the bijections connecting them to the forests of 1–2 trees and to permutations with the forbidden subsequences: 321, 4 1 523. In a similar way we describe the bijections between single-source animals on the triangular lattice and forests of binary trees. Each of these two objects presents a bijection with two permutations with the forbidden subsequences: 4132, 4231, 4312, 4321.