Arithmetic properties of odd ranks and k-marked odd Durfee symbols

Abstract

Let N0(m,n) be the number of odd Durfee symbols of n with odd rank m, and N0(a,M;n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We give explicit formulas for the generating functions of N0(a,M;n) and their ℓ-dissections where 0≤a≤M−1 and M,ℓ∈{2,4,8}. From these formulas, we obtain some interesting arithmetic properties of N0(a,M;n). Furthermore, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews (2007) conjectured that D20(n) is even if n≡4 or 6 (mod 8) and D30(n) is even if n≡1,9,11 or 13 (mod 16). Using our results on odd ranks, we prove Andrews' conjectures.