On a two-color partition series and its companions

Abstract: We study the two-color distinct-part series (S_1(q)), equivalently Andrews' generating function (v_d(q)) for strictly concave compositions, and its odd and even companions (T_o(q)) and (T_e(q)). We determine the coefficients of (S_1(q)) modulo (4) and obtain a complete criterion for the resulting Ramanujan-type progressions. For the even companion, we give a direct overpartition interpretation of its coefficients and show that two natural partition families are each counted by half of those coefficients. For the eta-normalized odd companion (C(q)=(q;q)_\infty T_o(q)), we prove a quintic self-similarity, derive exact vanishing relations and infinite sign changes for its coefficients, and show that (c(n)) can be nonzero only when (24n+28) is represented by (x^2+3y^2).