A Multiparameter Refinement of Euler’s Theorem

Euler’s partition theorem states that every integer has as many partitions into odd parts as into distinct parts. In this work, we reveal a new result behind this statement. On one hand, we study the partitions into odd parts according to the residue modulo 4 of the size of those parts occurring an odd number of times. On the other hand, we discuss the partitions into distinct parts with respect to the position of odd parts in the sequence. Some other statistics are also considered together, including the length, alternating sum and minimal odd excludant.