New Two-Line Arrays Representing Partitions

We present combinatorial interpretations for sums into two parameters from which we have, as special cases, combinatorial interpretations for many identities of Slater's list including Rogers-Ramanujan identities, unrestricted partitions, and Lebesgue's partition identity. In this work we are representing a number as a vector and providing representation of this vector as a sum of vectors. It is possible to write this representation as a two-line matrix which can be interpreted as lattice paths. We provide three distinct representations for unrestricted partitions. One of them has the property of giving a complete description for the conjugate partition.