  We prove an enumerative formula for the algebraic Euler characteristic of
Brill-Noether varieties, parametrizing degree d and rank r linear series on a
general genus g curve, with ramification profiles specified at up to two
general points. Up to sign, this Euler characteristic is the number of standard
set-valued tableaux of a certain skew shape with g labels. We use a flat
degeneration via the Eisenbud-Harris theory of limit linear series, relying on
moduli-theoretic advances of Osserman and Murray-Osserman; the count of
set-valued tableaux is an explicit enumeration of strata of this degeneration.
