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$$(x + x^2 + x^3 + x^4)^2 = x^2 (1 + x)^2 (1+x^2)^2 =[x + 2x^2 + x^3][x + 2x^3 + x^5].$$ corresponds to the re-labelling $1, 2, 2, 3$ and $1, 3, 3, 5$ .