Some Rogers-Ramanujan type partition theorems

Analogous to the celebrated Rogers-Ramanujan partition theorems, we obtain four partition theorems wherein the minimal difference for 4 about the first half of the parts of a partition (arranged in non-increasing order of magnitude) is 2. For example, we prove that the number of partitions of n, such that the minimal difference of the * first half of the summands' (that is, first [(/ + l)/2] summands in a partition into t summands) of any partition is 2, equals the number of partitions n into summands congruent to 1, 2, 5, 6, 8, 9 (mod 20).