# Complexity of counting MAXCUT in planar graphs — seemingly contradicting claims

Confusion is likely. Appears to me two papers give contradicting claims about the complexity of counting MAXCUT in planar graphs.

Exact Max 2-SAT: Easier and Faster p. 6

However, counting the number of Max 2-Sat or Max Cut solutions are $\#P$-complete even when restricted to planar graphs (results not explicitly stated but follow readily from results and reductions in [13, 20]).

[13] H. B. Hunt III, M. V. Marathe, V. Radhakrishnan, and R. E. Stearns, The complexity of planar counting problems, SIAM Journal of Computing 27 (1998), no. 4, 1142–1167.

[20] S. P. Vadhan, The complexity of counting in sparse, regular, and planar graphs, SIAM Journal of Computing 31 (2002), no. 2, 398–427.

Exponential Time Complexity of the Permanent and the Tutte Polynomial p. 13

Proposition 4.1. If #ETH holds, the coefficients of the polynomial w → Z(G; 2, w) for a given simple graph G cannot be computed in time exp(o(m)). Proof. The reduction is from #MaxCut and well-known ...Now, the coefficient of $(1 + w)^{m-c}$ in $Z(G; 2, w)$ is the number of cuts in $G$ of size $c$.

The Tutte polynomial on the hyperbola $(x-1)(y-1)=2$ is polynomially computable on planar graphs and (4) on the same page gives $Z(G;2,w)$ in terms of the Tutte polynomial on the hyperbola.

The first paper doesn't define MAXCUT, but assuming the usual definition, the two papers imply $P=\#P$.

What is wrong with this?