M. Merca

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math.CO2025

MacMahon-type $q$ -series

Mircea Merca

Motivated by earlier work of P.~A.~MacMahon and recent contributions of T.~Amdeberhan, G.~E.~Andrews, K.~Ono, A.~Singh, and R.~Tauraso on higher-order partition enumerants, we stud…

math.CO2025

Do perfect powers repel partition numbers?

Mircea Merca, Ken Ono, Wei-Lun Tsai

In 2013 Zhi-Wei Sun conjectured that $p (n)$ is never a power of an integer when $n > 1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the…

math.CO2024

Elementary symmetric partitions

Cristina Ballantine, George Beck, Mircea Merca +1

Let e_k(x_1,...,x_l) be an elementary symmetric polynomial and let mu = (mu_1,...,mu_l) be an integer partition. Define pre_k(mu) to be the partition whose parts are the summands i…

math.CO2024

Partitions and elementary symmetric polynomials -- an experimental approach

Cristina Ballantine, George Beck, Mircea Merca

Given a partition $I^»$ , we write $e_{j} (I^»)$ for the $j^{th}$ elementary symmetric polynomial $e_{j}$ evaluated at the parts of $I^»$ and $e_{j} p_{A} (n)$ for the sum of $e_{j} (I^»)$ a…

MacMahon-type q-series

Do perfect powers repel partition numbers?

Elementary symmetric partitions

Partitions and elementary symmetric polynomials -- an experimental approach

MacMahon-type $q$ -series