Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions |
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Authors: | Song Heng Chan Renrong Mao |
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Institution: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang link, Singapore, 637371, Republic of Singapore |
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Abstract: | We prove two monotonicity properties of N(m,n), the number of partitions of n with rank m. They are (i) for any nonnegative integers m and n, and, (ii) for any nonnegative integers m and n such that n?12, n≠m+2, G.E. Andrews, B. Kim, and the first author introduced ospt(n), a function counting the difference between the first positive rank and crank moments. They proved that ospt(n)>0. In another article, K. Bringmann and K. Mahlburg gave an asymptotic estimate for ospt(n). The two monotonicity properties for N(m,n) lead to stronger inequalities for ospt(n) that imply the asymptotic estimate. |
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Keywords: | Partitions Ranks Cranks Monotonicity Inequality |
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