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Fig. S1: Eight possible combinations from a source pool of three distinct species are enumerated with the binomial coefficient, C(n, k) = n!/(k!(n-k)!. Totals for the null set, 1-way, 2-way, and 3-way possibilities are, respectively, C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, C(3,3) = 1. This gives a total of eight (2R; 23 = 8).

Fig. S2: Enumeration of the possible combinations from random sampling of a moderately sized source pool of 15 distinct chemicals is depicted with a histogram. There is a total of 32,768 (215) unique k-way combinations. The binomial coefficient, C(n,k) = n!/(k!(n-k)!), computes the outcome for the possible k-way combinations; for example, there are 105 binary (two-way) combinations C(15,2) = 105.

Fig. S3: The West Indian Finch matrix (Gotelli and Abele, 1982), an example of a species presence--absence matrix used by ecologists.

Table S1: Comparison of the simulation of the WIF matrix with the two methods for null model 8.

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