A Self-Conjugate Partition Analog of $(t,t+1)$-Core Partitions with Distinct Parts
Huan Xiong, Lihong Yang
TL;DR
This work introduces a proper self-conjugate analogue \mathcal{DS}(t) of $(t,t+1)$-core partitions with distinct parts by focusing on self-conjugate partitions whose first Durfee-square parts are distinct. It develops an enumerative framework via the statistics $f(t)=|\mathcal{DS}(t)|$, $g(t)$ (sum of main-diagonal hook lengths), and $h(t)$ (sum of partition sizes), deriving recurrences $f(t)=f(t-1)+f(t-3)$, $g(t)=g(t-1)+g(t-3)+f(t-3)$, and $h(t)=2t\,g(t-1)+(2t-2)g(t-3)-h(t-1)-h(t-3)+f(t-3)$, with explicit generating functions $F(x)=1/(1-x-x^3)$, $G(x)=x^3F(x)/(1-x-x^3)$, and a formula for $H(x)$. The paper then determines the structure of maximum-size partitions in \mathcal{DS}(t): $2t-1$ must appear in $MD(\lambda)$, certain small diagonal hooks are excluded, and the maximal size $m(t)$ is given by a parity- and residue-based piecewise formula involving $t$ and $\left\lfloor\frac{2t+1}{12}\right\rfloor$, with a unique maximizing partition. Together, these results provide a coherent self-conjugate analogue to the classical $(t,t+1)$-core distinct-part theory and yield explicit counts and extremal sizes for this family.
Abstract
Simultaneous core partitions have been widely studied in the past 20 years. In 2013, Amdeberhan gave several conjectures on the number, the average size, and the largest size of $(t,t+1)$-core partitions with distinct parts, which was proved and generalized by Straub, Xiong, Nath-Sellers, Zaleski-Zeilberger, Paramonov, and many other mathematicians. In this paper, we introduce a proper self-conjugate partition analog of $(t,t+1)$-core partitions with distinct parts, and derive the number, the average size, and the largest size for such core partitions.