A comparison of Integer partitions based on smallest part

Damanvir Singh Binner¹ ; Amarpreet Rattan

¹Sant Longowal Institute of Engineering and Technology, Punjab, India

The electronic journal of combinatorics, Tome 31 (2024) no. 1

Résumé

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general $s$ and also find a precise nonnegativity result for the case $s=3$.

Détail

DOI : 10.37236/11978

Classification : 05A17, 05A20, 05A15, 11P81, 11P82
Mots-clés : partition inequalities, partitions with bounded differences between largest and smallest parts