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

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.

K-Knuth Equivalence for Increasing Tableaux

A $K$-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced by Buch, Kresch, Shimozono, Tamvakis, and Yong (2006) and Buch and Samuel (2013), respectively. The resulting $K$-Knuth equivalence relations on words and increasing tableaux on $[n]$ has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper, we give several new families of URTs and a bound on the length of intermediate words connecting two $K$-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are $K$-Knuth equivalent and to compute all $K$-Knuth equivalence classes of tableaux on $[n]$.

Linear time equivalence of Littlewood―Richardson coefficient symmetry maps

International audience Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the Littlewood―Richardson coefficient conjugation symmetry, i.e. $c_{\mu, \nu}^{\lambda} =c_{\mu^t,\nu^t}^{\lambda ^t}$. Tableau―switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon―Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schützenberger involution and reciprocally. Thus the Benkart―Sottile―Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schützenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo. Benkart, Sottile, et Stroomer ont complètement caractérisé par équivalence et équivalence duelle à Knuth une preuve bijective de la symétrie de la conjugaison des coefficients de Littlewood―Richardson, i.e. $c_{\mu, \nu}^{\lambda} =c_{\mu^t,\nu^t}^{\lambda ^t}$. Le tableau-switching donne un algorithme par produire une telle preuve bijective. Fulton a montré que les bijections de White et de Hanlon et Sundaram sont des versions de cette bijection. Dans ce papier on exhibe explicitement le mot de Yamanouchi produit par cette bijection de conjugaison lequel à son tour conduit à une nouvelle version très naturelle de la même bijection déjà considérée indépendamment. Une conséquence de cette dernière construction c'est qu'en utilisant des notions de Complexité Computationnelle Relative nous pouvons montrer que cette bijection de symétrie de la conjugaison est linéairement réductible à l'involution de Schützenberger et réciproquement. Ainsi la bijection de symétrie de la conjugaison de Benkart, Sottile et Stroomer avec les deux versions mentionnées, tout comme les trois versions de la bijection de la commutativité, et l'involution de Schützenberger sont linéairement réductibles les unes aux autres. Ça répond à une question posée par Pak et Vallejo.