The crank moments weighted by the parity of cranks |
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Authors: | Kathy Q Ji Alice X H Zhao |
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Institution: | 1.Center for Applied Mathematics,Tianjin University,Tianjin,People’s Republic of China;2.Center for Combinatorics, LPMC-TJKLC,Nankai University,Tianjin,People’s Republic of China |
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Abstract: | In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\). When \(k=0\), the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\), where \(M_e(n)\) (resp. \(M_o(n)\)) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\). |
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