A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $\frac{ n^2}{ 48}$ ($n$ even) or $\frac{ (n+3)^2}{ 48}$ ($n$ odd). We solve the analogous problem for $m$-gons (for arbitrary but fixed $m\geq3$), and for polygons (with arbitrary number of sides). We also show that the solution to the latter is asymptotic to $\frac{ 2^{ n-1}}n$, and the former (for fixed $m$) to $\frac{ 2^{ m-1}-m}{ 2^mm!}n^{ m-1}$.
Categories: math.CO math.GR