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By modifying the definition of moments of ranks and cranks, we study the odd moments of ranks and cranks. In particular, we prove the inequality between the first crank moment M 1 (n) and the first rank moment N 1 (n): M 1 (n) > N 1 (n). We also study new counting function ospt(n) which is equal to M 1 (n) − N 1 (n).

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