Some Elementary Partition Inequalities and Their Implications

2017/08/07 by Berkovich, Alexander, Uncu, Ali K.
#05A15 #05A17 #05A19 #05A20 #11B65 #11P81 #11P84 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1708.01957

Abstract

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for L≥ 1, the number of partitions with l-s ≤ L and s=1 is greater than the number of partitions with l-s≤ L and s>1. Here l and s are the largest part and the smallest part of the partition, respectively.

Related

• A conjecture about partitions
• Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity
• Counting Partitions on the Abacus
• Combinatorial proof of an inequality on some partitions separated by parity
• Gap between the largest and smallest parts of partitions and Berkovich and Uncu's conjectures