Generalized Robinson-Schensted correspondences: a new algorithm

The Electronic Journal of Combinatorics

We discuss the Robinson-Schensted and Schützenberger algorithms, and the fundamental identities they satisfy, systematically interpreting Young tableaux as chains in the Young lattice. We also derive a Robinson-Schensted algorithm for the hyperoctahedral groups. Finally we show how the mentioned identities imply some fundamental properties of Schützenberger's glissements.

Discrete Mathematics, 2005

In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson-Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided cell which have a minimal number of inversions is the set of permutations which have a maximal number of inversions in conjugated Young subgroups. We also give an interpretation of these sets with particular tableaux, called reading tableaux. As a corollary, we give the set of elements in a two-sided cell which have a maximal number of inversions.

Metrika, 2018

2008

The set of orbits of GL(V ) in F l(V ) × F l(V ) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V ) arising from F l(V )×F l(V )×V . We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V )× V .

Lecture Notes in Computer Science, 2016

A correspondence is a set of mappings that establishes a relation between the elements of two data structures (i.e. sets of points, strings, trees or graphs). If we consider several correspondences between the same two structures, one option to define a representative of them is through the generalised median correspondence. In general, the computation of the generalised median is an NPcomplete task. In this paper, we present two methods to calculate the generalised median correspondence of multiple correspondences. The first one obtains the optimal solution in cubic time, but it is restricted to the Hamming distance. The second one obtains a sub-optimal solution through an iterative approach, but does not have any restrictions with respect to the used distance. We compare both proposals in terms of the distance to the true generalised median and runtime.

HAL (Le Centre pour la Communication Scientifique Directe), 2022

A dissimilarity d on a set S of size n is said to be Robinson if its matrix can be symmetrically permuted so that its elements do not decrease when moving away from the main diagonal along any row or column. Equivalently, S admits a total order < such that i < j < k implies that d(i, j) ≤ d(i, k) and d(j, k) ≤ d(i, k). Intuitively, d is Robinson if S can be represented by points on a line. Recognizing Robinson dissimilarities has numerous applications in seriation and classification. Robinson dissimilarities also play an important role in the recognition of tractable cases for TSP. In this paper, we present two simple algorithms (inspired by Quicksort) to recognize Robinson dissimilarities. One of these algorithms runs in O(n 2 log n), the other one runs in O(n 3) in worst case and in O(n 2) on average.

2009

The set of orbits of GL(V) in Fl(V) × Fl(V) × V is finite, and is parametrized by the set of certain decorated permutations in a work of Magyar, Weyman, Zelevinsky. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of GL(V) arising from Fl(V) ×Fl(V) ×V. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on GL(V) × V.

BIT, 1980

A new proof of the Miihlbach-Neville-Aitken algorithm for interpolation by a linear family of functions forming a Chebyshev system is given. This proof is based on Sylvester's identity for determinants. The algorithm is then applied to the general interpolation problem, and applications to orthogonal polynomials and Pad6-type approximants are treated. Finally the extension to rational interpolation is also studied. I. Introduction. A general extrapolation algorithm including almost all the convergence acceleration methods actually known has been recently discovered by Hfivie [8] and Brezinski [3]. The question immediately arises if a general interpolation algorithm (i.e. an algorithm for interpolation by a linear family of functions forming a Chebyshev system which is similar to the Neville-Aitken scheme) exists or not. In fact such an algorithm has been obtained by Mfihlbach [13, 14] some years ago. The aim of this paper is twofold. First I shall present a new proof of the Miihlbach-Neville-Aitken algorithm (called the MNA-algorithm) similar to the proof I gave for the general extrapolation algorithm. This proof, which is shorter and, I think, simpler than the successive proofs given by MiJhlbach, is based on Sylvester's identity for determinants. Miihlbach's notations have been slightly changed to easier ones. The second aim of this paper is to show that the MNAalgorithm can be used for the general interpolation problem as described, for example, by Davis [6]. Applications to orthogonal polynomials and Pad6-type approximants are given. Finally the case of rational interpolation is studied. II. The Miihlbach-Neville-Aitken algorithm. Let (f~)i__>o be a family of functionsf~: G ~ K where G is an arbitrary set and K a commutative field of characteristic zero. The functions f~ will be assumed to form a Chebyshev system which means that all the Gram determinants Ifffxi)lij= 0 ..... k are different from zero whenever x o ..... x k are distinct points in G (for an extensive study of Chebyshev systems see the papers by MiJhlbach 1-13, 14] and the book by Karlin and Studden [9]).

Linear Algebra and its Applications, 2004

In correspondence analysis, rows and columns of a data matrix are depicted as points in low-dimensional space. The row and column profiles are approximated by minimizing the so-called weighted chisquared distance between the original profiles and their approximations, see or example, . In this paper, we will study the inverse correspondence analysis problem, that is, the possibilities of retrieving one or more data matrices from a low dimensional correspondence analysis solution. We will show that there exists a nonempty closed and bounded polyhedron of such matrices. We also present an algorithm to find the vertices of the polyhedron. A proof that the maximum of the Pearson chi-squared statistic is attained at one of the vertices is given. In addition, it is discussed how extra equality constraints on some elements of the data matrix can be imposed on the inverse correspondence analysis problem. As a special case, we present a method for imposing integer restrictions on the data matrix as well. The approach to inverse correspondence analysis followed here is similar to the one employed by De in their inverse multidimensional scaling problem.

Acta Informatica, 1993

We investigate the simplicity of solutions for instances of the Post Correspondence Problem, from the point of view of both index words and terminal words. This leads to the notion of a mixed primality type of an instance.