  There is an asymptotic relationship between the multiplicative relations
among multinomial coefficients and the (additive) recurrence property of
Shannon entropy known as the chain rule. We show that both types of identities
are manifestations of a unique algebraic construction: a $1$-cocycle condition
in \emph{information cohomology}, an algebraic invariant of phesheaves of
modules on \emph{information structures} (categories of observables). Baudot
and Bennequin introduced this cohomology and proved that Shannon entropy
represents the only nontrivial cohomology class in degree $1$ when the
coefficients are a natural presheaf of probabilistic functionals. The author
obtained later a $1$-parameter family of deformations of that presheaf, in such
a way that each Tsallis $\alpha$-entropy appears as the unique $1$-cocycle
associated to the parameter $\alpha$. In this article, we introduce a new
presheaf of \emph{combinatorial functionals}, which are measurable functions of
finite arrays of integers; these arrays represent \emph{histograms} associated
to random experiments. In this case, the only cohomology class in degree $0$ is
generated by the exponential function and $1$-cocycles are Fonten\'e-Ward
generalized multinomial coefficients. As a byproduct, we get a simple
combinatorial analogue of the fundamental equation of information theory that
characterizes the generalized binomial coefficients. The asymptotic
relationship mentioned above is extended to a correspondence between certain
generalized multinomial coefficients and any $\alpha$-entropy, that sheds new
light on the meaning of the chain rule and its deformations.
