We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrews in which he considers the generating function for partitions with respect to size, number of odd parts, and number of odd parts of the conjugate.
In this thesis, we present bijections proving partitions identities. In the first part, we generalize Dyson's definition of rank to partitions with successive Durfee squares. We then present two symmetries for this new rank which we prove using bijections generalizing conjugation and Dyson's map. Using these two symmetries we derive a version of Schur's… (More)
We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
The Robert-Bonamy formalism has been commonly used to calculate half-widths and shifts of spectral lines for decades. This formalism is based on several approximations. Among them, two have not been fully addressed: the isolated line approximation and the neglect of coupling between the translational and internal motions. Recently, we have shown that the… (More)
We present a generalization, which we call (k, m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Roger-Ramanujan identities. We… (More)