[Paper Review] Core partitions with distinct parts

Huan Xiong|arXiv (Cornell University)|Aug 31, 2015

Advanced Mathematical Identities21 references13 citations

TL;DR

This paper provides a complete solution to Amdeberhan's conjecture on $(t,t+1)$-core partitions with distinct parts, deriving the generating function for $t$-core partitions with distinct parts and proving exact formulas for their number, largest size, total size, and average size using combinatorial methods and Fibonacci identities.

ABSTRACT

Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core partitions with distinct parts is obtained. We also prove the results on the number, the largest size and the average size of $(t, t + 1)$-core partitions. This gives a complete answer to a conjecture of Amdeberhan, which is partly and independently proved by Straub, Nath and Sellers, and Zaleski recently.

Motivation & Objective

• To resolve Amdeberhan's 2015 conjecture on the enumeration and size statistics of $(t,t+1)$-core partitions with distinct parts.
• To derive the generating function for $t$-core partitions with distinct parts using $eta$-sets and combinatorial constraints.
• To establish exact formulas for the number, largest size, total sum of sizes, and average size of $(t,t+1)$-core partitions with distinct parts.
• To provide a self-contained proof of these results using recursive decomposition and Fibonacci number identities.
• To confirm and extend prior partial results by Straub, Nath and Sellers, and Zaleski on related core partition families.

Proposed method

• Utilizes $eta$-sets to encode $t$-core partitions with distinct parts, where each set $B$ satisfies $x_i x_{i+1} = 0$ for consecutive indices.
• Derives the generating function via a sum over admissible $(n_1, u_2, u_3, u_{t-1})$-tuples in $ u_t$, with exponent given by a quadratic form involving $i n_i + t inom{n_i}{2} - inom{ u}{2}$.
• Applies recursive decomposition based on whether $t-1$ is in the $eta$-set, leading to recurrence relations for the number and sum of sizes.
• Uses Fibonacci identities to prove that the total sum of sizes equals $\sum_{i+j+k=t+1, i,j,k\geq1} F_i F_j F_k$, where $F_i$ is the $i$-th Fibonacci number.
• Establishes the average size as the normalized sum $\sum_{i+j+k=t+1} \frac{F_i F_j F_k}{F_{t+1}}$.
• Verifies the recurrence $e_t = e_{t-1} + e_{t-2} + F_{t-1}$ for the total size, and confirms it matches the Fibonacci triple sum.

Experimental results

Research questions

• RQ1 What is the generating function for $t$-core partitions with distinct parts?
• RQ2 What is the exact number of $(t,t+1)$-core partitions with distinct parts?
• RQ3 What is the largest possible size of such partitions?
• RQ4 What is the total sum and average size of all $(t,t+1)$-core partitions with distinct parts?
• RQ5 How do the number and size statistics of these partitions relate to Fibonacci numbers?

Key findings

• The generating function for $t$-core partitions with distinct parts is $\sum_{n\geq 0} cd_t(n) q^n = \sum_{(n_1,\dots,n_{t-1}) \in \mathcal{C}_t} q^{\sum_{i=1}^{t-1} \left( i n_i + t \binom{n_i}{2} \right) - \binom{\sum n_i}{2}}$, where $\mathcal{C}_t$ is the set of tuples with no two consecutive nonzero entries.
• The number of $(t,t+1)$-core partitions with distinct parts is the Fibonacci number $F_{t+1}$.
• The largest size of such partitions is $\left\lfloor \frac{1}{3} \binom{t+1}{2} \right\rfloor$.
• The number of such partitions achieving the maximum size is 2 if $t \equiv 1 \pmod{3}$, and 1 otherwise.
• The total sum of the sizes of all $(t,t+1)$-core partitions with distinct parts is $\sum_{i+j+k=t+1, i,j,k\geq1} F_i F_j F_k$.
• The average size is $\sum_{i+j+k=t+1, i,j,k\geq1} \frac{F_i F_j F_k}{F_{t+1}}$.