Horiuti's equation relates four different ratios of this entity, and the (*) Saha equation gives its ratios between an ion plus electron and a neutral atom.  
Fisher and Kasteleyn developed an algorithm calculating this entity in planar graphs in polynomial time, though Jensen and Sinclair proved that calculating it in general is #p-complete.  Horiuti's equation relates four different ratios of this entity, and the (*) Saha equation gives its ratios between an ion plus electron and a neutral atom.  For a hydrogen atom it has a value of 2, and an identically-named entity in mathematics is defined as the number of ways of writing an integer as a sum of positive integers.  Free energy is proportional to its natural logarithm, as it is given by the weighted sum over different states of e to the negative energy over temperature.  For 10 points, name this function, which in statistical mechanics comes in "canonical" and "grand canonical" forms.
Fisher and Kasteleyn developed an algorithm calculating this entity in planar graphs in polynomial time, though Jensen and Sinclair proved that calculating it in general is #p-complete.  
{partition function} [accept {partition} after “function”] (3) [DW]