Intervals and factors in the Bruhat order

Combinatorics International audience In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related by a certain property of their reduced words. We show that the property does not hold when w is a decomposable permutation, and that the property always holds when w is the longest permutation.

Invariant Principal Order Ideals under Foata’s Transformation

Let $\Phi$ denote Foata's second fundamental transformation on permutations. For a permutation $\sigma$ in the symmetric group $S_n$, let $\widetilde{\Lambda}_{\sigma}=\{\pi\in S_n\colon\pi\leq_{w} \sigma\}$ be the principal order ideal generated by $\sigma$ in the weak order $\leq_{w}$. Björner and Wachs have shown that $\widetilde{\Lambda}_{\sigma}$ is invariant under $\Phi$ if and only if $\sigma$ is a 132-avoiding permutation. In this paper, we consider the invariance property of $\Phi$ on the principal order ideals ${\Lambda}_{\sigma}=\{\pi\in S_n\colon \pi\leq \sigma\}$ with respect to the Bruhat order $\leq$. We obtain a characterization of permutations $\sigma$ such that ${\Lambda}_{\sigma}$ are invariant under $\Phi$. We also consider the invariant principal order ideals with respect to the Bruhat order under Han's bijection $H$. We find that ${\Lambda}_{\sigma}$ is invariant under the bijection $H$ if and only if it is invariant under the transformation $\Phi$.

A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

Ordnungsprinzipien im Spermatidenkern von Gryllus domesticus L Hinweise auf Polytäne Flaschenbürsten-Chromosomen

Most of the chromatin in the spermatid nuclei of the cricket Gryllus domestricus, lies on the periphery of the nucleus. This zone is 0.2 —0.3 μ wide in young, round nuclei and 0.3 —0.4 μ in older spearshaped nuclei. In the young spear-shaped nuclei this zone begins to exhibit negative birefringence in relation to the radius of the nucleus. Thus, while the chromatin zone in young, round nuclei is entwined with irregularly wound threads (diam. 7 mμ), in the older spear-shaped nuclei one observes 20 mμ thick, hollow tubes with 7 mμ thick walls. These tubes lie approximately vertical to the nuclear membrane and originate from the screw-like wound threads of young, round nuclei as they strech. Later the stretching is corrected. An aggregation of the now parallel lying elements follows at once and leads during the continued shrinking of the nucleus to the formation of thick strands. Now. their length is about 0.4 μ. Some thick strands reach far into the center of the nucleus and are seen here as a loosely woven thread mesh. This central region of the nucleus is isotropic.The approximately 12 thousand radially oriented chromatin threads, each with a diameter of 20 mμ, are not attached directly to the nuclear membrane but to a non-membranous auxiliary structure which in longitudinal sections sometimes appears to be doubly contoured, lies several tens of mμ inward and is almost 20 mμ thick. Calculations indicate that this auxiliary structure consists most probably of the long axial threads of individual pachytene chromatids.A model of branched chromosomes is discussed which is believed to fit best the findings in cricket sperms. According to this model chromatin side chains are attached laterally to longitudinal axial threads which elements are thought upon due consideration to be morphologically polytenic in nature.The principal order of arrangement in spermatid nuclei should thus be sought in the manner of the attachment of the longitudinal threads to the nuclear membrane, a process which commences in an apical or terminal position and extends subsequently to the entire length of the threads.