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We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-d morphisms from a general genus-g, n-marked curve C to Pr, sending the marked points on C to specified general points in Pr, is equal to (r + 1)g for sufficiently large d. This computation may be rephrased as an intersection problem on Grassmannians, which has a combinatorial interpretation in terms of Young tableaux by the Littlewood-Richardson rule. We give a bijection, generalizing the RSK correspondence, between the tableaux in question and the (r+1)ary sequences of length g, and we explore our bijection’s combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which r = 1 and several marked points map to the same point in P1, the number of morphisms is still 2g for sufficiently large d. 

Plants are continuously exposed to beneficial and pathogenic microbes, but how plants recognize and respond to friends versus foes remains poorly understood. Here, we compared the molecular response of Arabidopsis thaliana independently challenged with a Fusarium oxysporum endophyte Fo47 versus a pathogen Fo5176. These two F. oxysporum strains share a core genome of about 46 Mb, in addition to 1,229 and 5,415 unique accessory genes. Metatranscriptomic data reveal a shared pattern of expression for most plant genes (about 80%) in responding to both fungal inoculums at all timepoints from 12 to 96 h postinoculation (HPI). However, the distinct responding genes depict transcriptional plasticity, as the pathogenic interaction activates plant stress responses and suppresses functions related to plant growth and development, while the endophytic interaction attenuates host immunity but activates plant nitrogen assimilation. The differences in reprogramming of the plant transcriptome are most obvious in 12 HPI, the earliest timepoint sampled, and are linked to accessory genes in both fungal genomes. Collectively, our results indicate that the A. thaliana and F. oxysporum interaction displays both transcriptome conservation and plasticity in the early stages of infection, providing insights into the fine-tuning of gene regulation underlying plant differential responses to fungal endophytes and pathogens. [Formula: see text] Copyright © 2021 The Author(s). This is an open access article distributed under the CC BY-NC-ND 4.0 International license .