Dissecting the Stanley partition function |
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Authors: | Alexander Berkovich Frank G Garvan |
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Institution: | Department of Mathematics, The University of Florida, Gainsville, FL 32611-8105, USA |
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Abstract: | Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let pi(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π′ is the conjugate of π. Stanley Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p0(n)-p2(n). Recently, Swisher The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show thatand that for sufficiently large n In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates. |
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Keywords: | Generating functions Stanley's partitions Even/odd dissection Upper bounds Asymptotic formulas Partition inequalities |
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