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We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SL r -webs, and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When r equals 2 or 3, it is a reformulation of the SL 2 -and SL 3 -web immanants defined by the second author. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of SL r -webs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity.

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$S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.

Let S be a surface, G a simply-connected classical group, and G ′ the associated adjoint form of the group. We show that the spaces of moduli spaces of framed local systems X G ′ ,S and A G,S , which were constructed by Fock and Goncharov ([FG1]), have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in [FG1], and also allows one to quantize higher Teichmuller space following the formalism of [FG2], [FG3], and [FG5], which was previously only possible when G had type A.