Families of parking functions counted by the schröder and baxter numbers

We define two new families of parking functions: one counted by Schröder numbers and the other by Baxter numbers. These families both include the well-known class of non-decreasing parking functions, which is counted by Catalan numbers and easily represented by Dyck paths, and they both are included in the class of underdiagonal sequences, which are bijective to permutations. We investigate their combinatorial properties exhibiting bijections between these two families and classes of lattice paths (Schröder paths and triples of non-intersecting lattice paths) and discovering a link between them and some classes of pattern avoiding permutations. Then, we provide a quite natural generalization for each of these families that results in some enumeration problems tackled by ECO method

Clustering of surface NMDA receptors is mainly mediated by the C-terminus of GluN2A in cultured rat hippocampal neurons

N-methyl-D-aspartate receptors (NMDARs) containing different GluN2 subunits play distinct roles in synaptic plasticity. Such differences may not only be determined by the channel properties, but also by differential surface distribution and synaptic localization. In the present study, using a Cy3-conjugated Fab fragment of the GFP antibody to label surface-located GluN2 subunits tagged with GFP at the N-terminus, we observed the membrane distribution patterns of GluN2A- or GluN2B-containing NMDARs in cultured rat hippocampal neurons. We found that surface NMDARs containing GluN2A, but not those containing GluN2B, were inclined to cluster at DIV7. Swapping the carboxyl termini of the GluN2 subunits completely reversed these distribution patterns. In addition, surface NMDARs containing GluN2A were preferentially associated with PSD-95. Taken together, the results of our study suggest that the clustering distribution of GluN2A-containing NMDARs is determined by the GluN2A C-terminus, and its interaction with PSD-95 plays an important role in this process.open