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Generating function in Maple format
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First 15 coefficients

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Av(1234, 1243, 1324, 1342) Av(1234, 1243, 1324, 1432) Av(1234, 1243, 1324, 2134) Av(1234, 1243, 1324, 2143) Av(1234, 1243, 1324, 2314) Av(1234, 1243, 1324, 2341) Av(1234, 1243, 1324, 2413) Av(1234, 1243, 1324, 2431) Av(1234, 1243, 1324, 3214) Av(1234, 1243, 1324, 3241) Av(1234, 1243, 1324, 3412) Av(1234, 1243, 1324, 3421) Av(1234, 1243, 1324, 4231) Av(1234, 1243, 1324, 4321) Av(1234, 1243, 1342, 1423) Av(1234, 1243, 1342, 1432) Av(1234, 1243, 1342, 2134) Av(1234, 1243, 1342, 2143) Av(1234, 1243, 1342, 2314) Av(1234, 1243, 1342, 2341) Av(1234, 1243, 1342, 2413) Av(1234, 1243, 1342, 2431) Av(1234, 1243, 1342, 3124) Av(1234, 1243, 1342, 3142) Av(1234, 1243, 1342, 3214)