Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo 10, and additional results modulo 6 and 10 for the M2 rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the M2 rank modulo 10 using elementary methods.Furthermore, we will make use of the following notation of Mao. 1 For positive integers a < b, defineLemma 2.1 (Mao [7]). Given positive integers a < b, the q-series coefficients of L a,b are all nonnegative.Mao proved rank difference formulas that we will use in our proof of Theorem 1.3. First, for unrestricted partitions, Mao proved the following theorem. Theorem 2.2 (Mao [7]). We have that ∞ n=0 N (0, 10, n) + N (1, 10, n) − N (4, 10, n) − N (5, 10, n) q n = J 25 J 5 50 J 2 20,50 J 4 10,50 J 3 15,50 + 1 J 25 ∞ n=−∞ (−1) n q 75n(n+1)/2+5 1 + q 25n+5 + q J 25 J 5 50 J 5,50 J 2 10,50 J 2 15,50 + q 2 J 25 J 5 50 J 2 5,50 J 15,50 J 2 + q 2 J 25 J 5 50 J 20,50 J 3 10,50 J 3 15,50 + q 3 J 25 J 5 50 J 5,50 J 10,50 J 2 15,50 J 20,50 + q 4 J 25 J 5 50 J 2 20,50 J 25,50 2q 5 J 4 10,50 J 4 1 We note that our definition of L a,b differs from Mao's in that the roles of a and b are reversed.3 = 2q 5 J 15 100 J 10,100 J 50,100 J 3 5,100 J 2 15,100 J 2 20,100 J 3 25,100 J 30,100 J 2 35,100 J 3 45,100 + 1 J 25,100 ∞ n=−∞ (−1) n q 50n 2 +25n 1 + q 50n+10 + q J 15 100We see that S, T 1 , . . . , T 4 all have nonnegative coefficients. Thus to prove (3), it suffices to show that J 2 5 J 5 10 J 2 4,10 J 3 3,10 J 4 2,10has positive coefficients. Let T 1 + T 2 + T 3 + T 4 = ∞ n=1 a(n)q n , and let J 2 5 J 5 10 J 2 4,10 J 3 3,10 J 4 2,10 = 1 + ∞ n=1 b(n)q n .We will show that b(n) > a(n) for all n ≥ 1. 5 = 1 J 5,20
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