Singular Overpartitions and Partitions with Prescribed Hook Differences

Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with 1 ≤ i < K/2. The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define (K, i, α, β)-singular overpartitions for positive integers α, β with α+β<K, and then we show some connections between such singular overpartitions and ordinary partitions.