Structural properties of Potts model partition functions and chromatic polynomials for lattice strips

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width Ly and arbitrary length Lx has the form Z(G,q,v)=∑j=1NZ,G,λcZ,G,j(λZ,G,j)Lx, where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet (v=−1) is the chromatic polynomial P(G,q). Using coloring and transfer matrix methods, we give general formulas for CX,G=∑j=1NX,G,λcX,G,j for X=Z,P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient cZ,G,j of degree d in q: c(d)=U2d(q/2), where Un(x) is the Chebyshev polynomial of the second kind, we determine the number of λZ,G,j's with coefficient c(d) in Z(G,q,v) for these cyclic strips of width Ly to be nZ(Ly,d)=(2d+1)(Ly+d+1)−12LyLy−d for 0⩽d⩽Ly and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λZ,G,j is calculated to be NZ,Ly,λ=2LyLy. Results are also presented for the analogous numbers nP(Ly,d) and NP,Ly,λ for P(G,q). We find that nP(Ly,0)=nP(Ly−1,1)=MLy−1 (Motzkin number), nZ(Ly,0)=CLy (the Catalan number), and give an exact expression for NP,Ly,λ. Our results for NZ,Ly,λ and NP,Ly,λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations NZ,Ly,λ=2NDA,tri,Ly and NP,Ly,λ=2NDA,sq,Ly, where NDA,Λ,n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths NZ,Ly,λ∼Ly−1/24Ly and NP,Ly,λ∼Ly−1/23Ly as Ly→∞. Some general geometric identities for Potts model partition functions are also presented.