Yee’s Bijective Proof of Bousquet-Mélou And Eriksson’s Refinement of the Lecture Hall Partition Theorem

Beril Talay
Mathematics, MSc. Thesis, 2018

Thesis Jury

Assoc.Prof Kağan Kurşungöz (Thesis Advisor), Asst. Prof. Dr. Ayesha Asloob Qureshi,

Asst. Prof. Dr. Zafeirakis Zafeirakopoulos

Date & Time: 30th, 2018 –  12:00

Place: FENS 2008

Keywords : integer partition, lecture hall partitions, partition bijection, partition analysis

Abstract

A partition λ=(λ_1,λ_2,…,λ_n) of a positive integer N is a lecture hall partition of length n if it satisfies the condition 0≤ λ_1/1 ≤ λ_2/2 ≤⋯≤ λ_n/n.

Lecture hall partitions are introduced by Bousquet-Mélou and Eriksson, while studying Coxeter groups and their Poincaré series. Bousquet-Mélou and Eriksson showed that the number of lecture hall partitions of length n where the alternating sum of the parts is k equals to the number of partitions into k odd parts which are less than 2n by a combinatorial bijection. Then, Yee also proved the fact by combinatorial bijection which is differently defined for of the one of the bijections that suggested by Bousquet-Mélou and Eriksson. In this thesis we give Yee’s proof with details and further possible problems which arises from a paper of Corteel et al.