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math.CO2024
Distribution of hooks in self-conjugate partitions
William Craig, Ken Ono, Ajit Singh
We confirm the speculation that the distribution of -hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the num…
math.CO2024
Remarks on MacMahon's -series
Ken Ono, Ajit Singh
In his important 1920 paper on partitions, MacMahon defined the partition generating functions \begin{align*} A_k(q)=\sum_{n=1}^{\infty}\mathfrak{m}(k;n)q^n&:=\sum_{0< s_1<s_2<\cdo…
math.CO2023
Hook lengths in self-conjugate partitions
Tewodros Amdeberhan, George E. Andrews, Ken Ono +1
In 2010, G.-N. Han obtained the generating function for the number of size hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitio…