Unimodality of k-Regular Partitions into Distinct Parts with Bounded Largest Part

2023/06/07 by Janet J. W. Dong, Dong, Janet J. W., Kathy Q. Ji +1
Mathematics · #Advanced Combinatorial Mathematics #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2306.04438

openalex publication_date 2023/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A k-regular partition into distinct parts is a partition into distinct parts with no part divisible by k. In this paper, we provide a general method to establish the unimodality of k-regular partition into distinct parts where the largest part is at most km+k-1. Let d_k,m(n) denote the number of k-regular partition of n into distinct parts where the largest part is at most km+k-1. In line with this method, we show that d_4,m(n)≥ d_4,m(n-1) for m≥ 0, 1≤ n≤ 3(m+1)² and n≠ 4 and d_8,m(n)≥ d_8,m(n-1) for m≥ 2 and 1≤ n≤ 14(m+1)². When 5≤ k≤ 10 and k≠ 8, we show that d_k,m(n)≥ d_k,m(n-1) for m≥ 0 and 1≤ n≤ \lfloor(k(k-1)(m+1)²)/(4)\rfloor.

Related

• Proof of a conjecture of Stanley-Zanello
• Multiplicative properties of the number of k-regular partitions
• Partitions with unique largest part and their generating functions
• The number of maximal unrefinable partitions
• Identities involving partitions with distinct even parts and 4-regular partitions