While Shannon entropy is related to the growth rate of multinomial coefficients, we show that Tsallis 2-entropy is connected to their $q$-version; when $q$ is a prime power, these coefficients count the number of flags in $\mathbb{F}_q^n$ with prescribed length and dimensions ($\mathbb{F}_q$ denotes the field of order $q$). In particular, the $q$-binomial coefficients count vector subspaces of given dimension. We obtain this way a combinatorial explanation for non-additivity. 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 distribution, that generates at time $n$ a vector subspace of $\mathbb{F}_q^n$. The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. We discuss the applications to information theor 查看全文>>