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Abstract: |
In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N^0 (m,n) denote the number of odd Durfee symbols of n with odd rank m, and N^0 (r,m;n) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N^0?(r,12;n) by utilizing some identities involving Appell-Lerch sums m(x,q,z) and a universal mock theta function g(x,q). Based on these formulas, we determine the signs of N^0 (r,12;4n+t)-N^0 (s,12;4n+t) for all 0≤r,s≤6 and 0≤t ≤3. In particular, we prove that N^0 (2,12;4n+1)=N^0 (4,12;4n+1). Moreover, let D_k^0 (n) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D_2^0 (8n+s) and D_3^0 (16n+t) are even with s∈\{4,6\} and t∈\{1,9,11,13\} which were confirmed by Wang. In this paper, we found new congruences for D_k^0 (n). In particular, for k=2 or 3, we give characterizations of n such that D_k^0 (n) is odd and prove that D_k^0 (n) take even values with probability 1 for n≥0.
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