THE SEXTUPLE COMPLETE PARTITIONS OF INTEGERS

This paper presents the concepts of sextuple (6 – tuple) complete partitions of integers and an attempt has been given for the theorem based on the last part of sextuple complete partitions of integers

Geometry and complexity of O'Hara's algorithm

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.Comment: 20 pages, 4 figure

Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

In 1958, Richard Guy proved that the number of partitions of n into odd parts greater than one equals the number of partitions of n into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.