Gordon's Partition Theorem

Let  denote the number of partitions into  parts not congruent to 0, , or  (mod ). Let  denote the number of partitions of  wherein

1. 1 appears as a part at most  times.

2. The total number of appearances of  and  (i.e., any two consecutive integers) together is at most .

Then Gordon's partition theorem states that for ,

The first Rogers-Ramanujan identity corresponds to , and the second to , .

REFERENCES:

Andrews, G. E. and Santos, J. P. O. "Rogers-Ramanujan Type Identities for Partitions with Attached Odd Parts." Ramanujan J. 1, 91-99, 1997.

Gordon, B. "A Combinatorial Generalization of the Rogers-Ramanujan Identities." Amer. J. Math. 83, 393-399, 1961.