Some inequalities between $M(a,b,c;L;n)$ and the partition function $p(n)$

He, Bing  ; Li, Linpei ; Cao, Jian 

Canadian journal of mathematics, Tome 77 (2025) no. 6, pp. 2148-2170

Résumé

Let $p(n)$ and $M(m,L;n)$ be the number of partitions of n and the number of partitions of n with crank congruent to m modulo L, respectively, and let $$ \begin{align*}M(a,b,c;L;n):= M(a,L;n) + M(b,L;n) + M(c,L;n).\end{align*} $$In this paper, we focus on some relations between $M(m,L;n)$ and $p(n),$ which Dyson, Andrews, and Garvan etc. have studied previously. By applying a modification of the circle method to estimate the Fourier coefficients of generating functions, we establish the following inequalities between $M(a,b,c;L;n)$ and $p(n):$ for $n \geq 467$, $$ \begin{align*} M(0,1,1;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(0,1,1;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n) < \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 0,1,5,8 ~ (\mathrm{mod}~9), \\ M(2,3,4;9;n)> \frac{p(n)}{3} &\;\: \mathrm{when}\;\: n \equiv 2,3,4,6,7 ~ (\mathrm{mod}~9). \end{align*} $$In the proof of these inequalities, an inequality for the logarithm of the generating function for $p(n)$ is derived and applied. Our method reduces the last possible counterexamples to $467 \leq n \leq 22471$, and it will produce more effective estimates when proving inequalities of such types.

Détail

DOI : 10.4153/S0008414X24000683

Mots-clés : Inequality, partition, crank, M(a, b, c;L;n), p(n), circle method