Log-Concavity of Infinite Product and Infinite Sum Generating Functions (2302.13327v1)

Abstract: We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${g_d(n)}{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate double sequences $\left{ p^{g_{d} }\left( n\right) \right}$ and $\left{ q^{g_{d} }\left( n\right) \right} $, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} \mu(\ell) \, g_d(n/\ell) }{n} }, \ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*} These coefficients are related to the number of partitions $\mathrm{p}\left( n\right) = p^{\sigma {1 }}\left ( n\right) $, plane partitions $pp\left( n\right) = p^{\sigma _{2 }}\left( n\right) $ of $n$, and Fibonacci numbers $F{2n} = q^{\psi _{1 }}\left( n\right) $. Let $n \geq 3$ and let $n \equiv 0 \pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n \equiv 2 \pmod{3}$. Let $n\equiv 1 \pmod{3}$. Then the log-concave property flips for almost all $d$.