Investigation of properties of equivalence classes of permutations by inverse Robinson — Schensted — Knuth transformation

Abstract

Introduction:All information about a permutation, i.e. about an element of a symmetric groupS(n), is contained in a pair of Young tableaux mapped to it by RSK transformation. However, when considering an infinite sequence of natural or real numbers instead of a permutation, all information about it is contained only in an insertion infinite Young tableau. The connection between the first element of an infinite sequence of uniformly distributed random values and the limit angle of the recording tableau nerve was found in a recent work by D. Romik and P. Śniady. However, so far there were no massive numerical experiments devoted to the reconstruction of the beginning of such a sequence by the beginning of an insertion Young tableau. The reconstruction accuracy is very important, because even the value of the first element of a sequence can be determined only by an infinite tableau.Purpose:Developing a software package for operations on Young diagrams and Young tableaux, and its application for numerical experiments with large Young tableaux. Studying the properties of Knuth equivalence classes and dual Knuth equivalence classes on a set of permutations by numerical experiments using direct and inverse RSK transformation.Results:A software package is developed using the C ++ programming language. It includes functions for dealing with Young diagrams and tableaux. The dependence of values of the first element of a permutation obtained by inverse RSK transformation on the recording tableau nerve end coordinates was investigated by conducting massive numerical experiments. Standard deviations of these values were calculated for permutations of different sizes. We determined possible positions of 1 in permutations of the same Knuth equivalence class. It has been found out that the number of these positions does not exceed the number of corner boxes of the corresponding Young diagram. Experiments showed that for a fixed insertion tableau, the value of the first element of a permutation depends only on the recording tableau nerve end coordinates.

Fluctuations of Schensted row insertion

“…Remark 1.6. It would be interesting to compare Conjecture 1.5 to the results of some Monte Carlo experiments by Vassiliev, Duzhin, and Kuzmin [VDK19]. Note, however, that some of their results seem to be based on corrupted data.…”

On the convergence of bumping routes to their limit shapes in the RSK algorithm: numerical experiments

“…The problem of studying the trajectories of bumping routes in the RSK algorithm was first formulated in [6]. Later, the dynamics of motion of an element in the RSK algorithm and the dynamics of tableau P changes began to be studied from the combinatorial point of view in [7][8][9][10][11][12][13]. It led to the introduction of a bumping tree and a bumping forest [14], as well as the study of their combinatorial structure.…”