  There are two seemingly unrelated ideals associated with a simplicial complex
\Delta. One is the Stanley-Reisner ideal I_\Delta, the monomial ideal generated
by minimal non-faces of \Delta, well-known in combinatorial commutative
algebra. The other is the toric ideal I_{M(\Delta)} of the facet subring of
\Delta, whose generators give a Markov basis for the hierarchical model defined
by \Delta, playing a prominent role in algebraic statistics.
  In this note we show that the complexity of the generators of I_{M(\Delta)}
is determined by the Betti numbers of I_\Delta. The unexpected connection
between the syzygies of the Stanley-Reisner ideal and degrees of minimal
generators of the toric ideal provide a framework for further exploration of
the connection between the model and its many relatives in algebra and
combinatorics.
