conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions.has the chromatic polynomial t(t − 1) m−1 then the graph is a tree on m vertices. These results can be found in the introductory article [16].Richard P. Stanley [20] introduced an invariant called the chromatic symmetric function, which is stronger than the chromatic polynomial. For a finite simple graph G, the chromatic symmetric function X(G, x) of G is the formal power series

defined the chromatic symmetric function of a simple graph and has conjectured that every tree is determined by its chromatic symmetric function. Recently, Takahiro Hasebe and the author proved that the order quasisymmetric functions, which are analogs of the chromatic symmetric functions, distinguish rooted trees. In this paper, using a similar method, we prove that the chromatic symmetric functions distinguish trivially perfect graphs. Moreover, we also prove that claw-free cographs, that is, {K 1,3 , P 4 }-free graphs belong to a known class of e-positive graphs.

A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type A ℓ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type B ℓ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type A ℓ−1 and type B ℓ . In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type B ℓ under certain assumption.

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties.

The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free.Comment: We added Section 3 (v2). We corrected the proof of Theorem 1.3 and several typos (v3