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  1. Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–MartĂ­n–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $K$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux. 
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