In [BabStein] Babson and Steingŕımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There eithe...

In [BabStein] Babson and Steingŕımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There eithe...

Lauren Williams (joint work with Einar Steingrímsson) We introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the " Le-diagrams " of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe tha...

The linear code equivalence problem is to decide whether two linear codes over Fq are identical up to a linear isometry of the Hamming space. The support splitting algorithm [23] runs in polynomial time for all but a negligible proportion of all linear codes, and solves the latter problem by recovering the isometry when it is just a permutation of the code support. While for a binary alphabet i...

This paper presents new, carefully designed algorithms for five biobjective permutation flow shop scheduling problems that arise from the pairwise combinations of the objectives (i) makespan, (ii) the sum of the completion times of the jobs, and (iii) both, the weighted and nonweighted total tardiness of all jobs. The proposed algorithm combines two search methods, two-phase local search and Pa...

Makespan minimization in permutation flow-shop scheduling is a well-known hard combinatorial optimization problem. Among the 120 standard benchmark instances proposed by E. Taillard 1993, 23 have remained unsolved for almost three decades. In this paper, we present our attempts to solve these optimality using parallel Branch-and-Bound (BB) on GPU-accelerated Jean Zay supercomputer. We report ex...

We consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper–bound m[k](q) ≤ (k−1)!(q(q−1))/k for char(Fq) > e(k−3)/e and the lower–bound m[k](q) ≥ φ(k)(q(q−1))/k for q ≡ 1 (mod k). This is done by establishing a connection with the Fq–solutions of ...

Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bijective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and g...