Ricardo Mamede

ON THE LARGEST SIZE OF AN ANTICHAIN IN THE BRUHAT ORDER FOR A (2k,k)

We discuss a problem proposed by Brualdi and Deaett on the largest size of an antichain in the Bruhat order for the interesting combinatorial class of binary matrices of .

On the Largest Size of an Antichain in the Bruhat Order for Open image in new window

We discuss a problem proposed by Brualdi and Deaett on the largest size of an antichain in the Bruhat order for the interesting combinatorial class of binary matrices of .

Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

We present combinatorial Gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n.

Gray codes for signed involutions

Abstract In this paper we present two cyclic Gray codes for signed involutions. The first one has a natural construction, implemented by a CAT algorithm, based in the recursive formula for the number of signed involutions. The second code, although with a higher computational cost, has the smallest possible Hamming distance for this family of objects.

Gray codes for noncrossing and nonnesting partitions of classical types

In this paper, we present Gray codes for the sets of noncrossing partitions associated with the classical Weyl groups, and for the set of nonnesting partitions of type B. An algorithm for the generation of type D nonnesting partitions is developed in which a Gray code is given for those partitions having a zero-block, while the remaining are arranged in lexicographic order.

SINGLETON FREE SET PARTITIONS AVOIDING A 3-ELEMENT SET

The definition and study of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations, begun with Klazar. Sagan continued his work by considering set partitions which avoid a single partition of three elements, and Goyt generalized these results by considering partitions which avoid any family of partitions of a 3-element set. In this paper we enumerate and describe set partitions, even set partitions and odd set partitions without singletons which avoid any family of partitions of a 3-element set. The characterizations of these families allow us to conclude that the corresponding sequences are P-recursive. We also construct Gray codes for the sets of singletons free partitions that avoid a single partition of three elements.

Chains and Antichains in the Bruhat Order for Classes of (0, 1)-Matrices

Let \({\mathscr {A}}(R,S)\) denote the set of all matrices of zeros and ones with row sum vector R and column sum vector S. This set can be ordered by a generalization of the usual Bruhat order for permutations. Contrary to the classical Bruhat order on permutations, where permutations can be seen as permutation matrices, the Bruhat order on the class \({\mathscr {A}}(R,S)\) is not, in general, graded, and an interesting problem is the determination of bounds for the maximal length of chains and antichains in this poset. In this survey we aim to provide a self-contained account of the recent developments involving the determination of maximum lengths of chains and antichains in the Bruhat order on some classes of matrices in \({\mathscr {A}}(R,S)\).