Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type BB

International audienceAlignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableaux of type $B$.De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type $B$ correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type $B$ correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type $B$ correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type $B$ en termes de tableaux de permutations de type $B$

Positivity of chromatic symmetric functions associated with Hessenberg functions of bounce number 3

We give a proof of Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.Comment: 32 page

Fast Parameterless Ballistic Launch Point Estimation based on k-NN Search

This paper discusses the problem of estimating a ballistic trajectory and the launch point by using a trajectory similarity search in a database. The major difficulty of this problem is that estimation accuracy is guaranteed only when an identical trajectory exists in the trajectory database (TD). Hence, the TD must comprise an impractically great number of trajectories from various launch points. Authors proposed a simplified trajectory database with a single launch point and a trajectory similarity search algorithm that decomposes trajectory similarity into velocity and position components. These similarities are applied k-NN estimation. Furthermore, they used the iDistance technique to partition the data space of the high-dimensional database for an efficient k-NN search. Authors proved the effectiveness of the proposed algorithm by experiment.Defence Science Journal, Vol. 64, No. 1, January 2014, DOI:10.14429/dsj.64.295

Permutation module decomposition of the second cohomology of a regular semisimple Hessenberg variety

Regular semisimple Hessenberg varieties admit actions of associated Weyl groups on their cohomology space of each degree. In this paper, we consider the module structure of the cohomology spaces of regular semisimple Hessenberg varieties of type $A$. We define a subset of the Bialynicki-Birula basis of the cohomology space so that they become a module generator set of the cohomology module of each degree. We then use those generators to construct permutation submodules of the degree two cohomology module and show that they form a permutation module decomposition. Our construction is consistent with a known combinatorial result by Chow on chromatic quasisymmetric functions