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Generating function in Maple format
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First 15 coefficients

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Av(1243, 1324, 2143) Av(1324, 1342, 3142) Av(1324, 1432, 2431) Av(1342, 1432, 2431) Av(1342, 2341, 3142) Av(1342, 3142, 3241)