  Whereas Shannon entropy is related to the growth rate of multinomial
coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is
connected to their $q$-deformation; when $q$ is a prime power, these
$q$-multinomial coefficients count flags of finite vector spaces with
prescribed length and dimensions. In particular, the $q$-binomial coefficients
count vector subspaces of given dimension. We obtain this way a combinatorial
explanation for the nonadditivity of the quadratic entropy, which arises from a
recursive counting of flags. We show that statistical systems whose
configurations are described by flags provide a frequentist justification for
the maximum entropy principle with Tsallis statistics. We introduce then a
discrete-time stochastic process associated to the $q$-binomial probability
distribution, that generates at time $n$ a vector subspace of $\mathbb{F}_q^n$
(here $\mathbb{F}_q$ is the finite field of order $q$). The concentration of
measure on certain "typical subspaces" allows us to extend the asymptotic
equipartition property to this setting. The size of the typical set is
quantified by the quadratic entropy. We discuss the applications to Shannon
theory, particularly to source coding, when messages correspond to vector
spaces.
