A new bijective proof of a partition theorem of K. Alladi

Abstract

The theorem " the number of partitions of a positive integer n into distinct odd parts equals the number of partitions of n into parts = 2 and differing by 6 with strict inequality if a part is even" was first discovered and then proved by Krishnaswamy Alladi in the year 1999. The object of this paper is to give a different bijective proof of this result.

A bijective proof and generalization of Siladić’s Theorem

“…We build our bijection for Theorem 1.3 in the spirit of the bijective proof of the partition theorem of K. Alladi [1] given by Padmavathamma, R. Raghavendra and B. M. Chandrashekara [10]. The idea was introduced by Bressoud [3] in his bijective proof of Schur's theorem.…”

A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem

“…The objective of this paper is to provide a bijective proof of Theorem 1.3. Our proof is in the spirit of the combinatorial proof of Alladi's partition theorem [2,Theorem 1] given by Padmavathamma, Raghavendra and Chandrashekara [14].…”

“…Andrews [9] asked for a proof which would offer more insights into the refinement (1.1) of Göllnitz's theorem. There has been a lot of progress towards this direction, see [1,5,14]. The first combinatorial approach to Theorem 1.2 was provided by Alladi [1].…”

“…However, as mentioned by Alladi [1], his construction can not be used to give a bijection between the sets of partitions of n counted by B(n) and C(n). Padmavathamma, Raghavendra and Chandrashekara [14] presented a bijective proof of another partition theorem due to Alladi [2,Theorem 1], and remarked that their bijection also implies Theorem 1.2. They also noted that their method is very similar in spirit to Bressoud's [11] combinatorial proof of Schur's partition theorem.…”

Generalized bijective proof of the partition identity of M.V. Subbarao

“…Padmavathamma et al [4], [5] have given bijective proofs of the partition identities stated in Theorem 4 for the cases m = 2 and r = 2,3.…”