[Paper Review] On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts

Anthony Zaleski, Doron Zeilberger|arXiv (Cornell University)|Dec 28, 2017

Advanced Combinatorial Mathematics1 references3 citations

TL;DR

This paper investigates the enumeration of $(n+1,n+2)$-core partitions into odd parts, a long-standing open problem in partition theory. Using order ideals on a triangular lattice and advanced generating function techniques in Maple, the authors compute 23 terms of the sequence and provide strong evidence for an algebraic generating function, later confirmed via Paul Johnson's discovery of a deep connection to even-part partitions, enabling a fast algorithm and a conjectured algebraic equation for the generating function.

ABSTRACT

Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core partitions into odd parts remains elusive. Straub computed the first eleven terms of that sequence, and asked for a "formula," or at least a fast way, to compute many terms. While we are unable to find a "fast" algorithm, we did manage to find a "faster" algorithm, which enabled us to compute 23 terms of this intriguing sequence. We strongly believe that this sequence has an algebraic generating function, since a "sister sequence" (see the article), is OEIS sequence A047749 that does have an algebraic generating function. One of us (DZ) is pledging a donation of 100 dollars to the OEIS, in honor of the first person to generate sufficiently many terms to conjecture (and prove non-rigorously) an algebraic equation for the generating function of this sequence, and another 100 dollars for a rigorous proof of that conjecture. Finally, we also develop algorithms that find explicit generating functions for other, more tractable, families of (n+1,n+2)-core partitions.

Motivation & Objective

• To compute more terms of the sequence counting $(n+1,n+2)$-core partitions into odd parts, which had only 11 terms previously computed by Straub.
• To determine whether the generating function for this sequence is algebraic, based on strong numerical evidence and structural analogies.
• To develop algorithmic methods using order ideals and generating functions to enumerate restricted families of $(n+1,n+2)$-core partitions.
• To establish a connection between the odd-part and even-part variants of $(n+1,n+2)$-core partitions, leading to a fast algorithm for the odd-part sequence.

Proposed method

• The authors model $(n+1,n+2)$-core partitions as order ideals in a triangular lattice $A_n$, identified with the set $P_{n+1,n+2} = \mathbb{N} \setminus ((n+1)\mathbb{N} + (n+2)\mathbb{N})$.
• They classify order ideals based on their 'type'—defined by the parities of the largest and smallest labels in each diagonal—enabling recursive decomposition of the generating function.
• For fixed $k$, they restrict attention to order ideals confined to the $k$ outermost diagonals, which allows finite-state machine-like recursion and algebraic generating functions.
• Using the Maple packages `OddArmin.txt`, `core.txt`, and `stCorePlus.txt`, they implement algorithms to compute generating functions for restricted families, including those with bounded repetition or odd parts.
• They derive explicit rational generating functions for $k$-bounded repetition (e.g., $k=2,3,4$), showing these satisfy algebraic equations.
• They leverage Paul Johnson’s discovery that the odd-part sequence is related to the even-part sequence, which allows fast computation and conjectures an algebraic generating function.

Experimental results

Research questions

• RQ1 Is the generating function for the number of $(n+1,n+2)$-core partitions into odd parts algebraic, as strongly suggested by numerical evidence and analogy with related sequences?
• RQ2 Can a fast algorithm be constructed to compute many terms of this sequence, given the difficulty of direct enumeration?
• RQ3 What structural relationship exists between the sequences counting $(n+1,n+2)$-core partitions into odd parts and into even parts?
• RQ4 Can the generating function for the odd-part sequence be derived from the even-part sequence via a known combinatorial transformation?
• RQ5 What is the minimal number of terms required to conjecture the algebraic equation of the generating function?

Key findings

• The authors computed 23 terms of the sequence counting $(n+1,n+2)$-core partitions into odd parts, significantly extending the previously known 11 terms.
• The sequence is strongly conjectured to have an algebraic generating function, based on numerical evidence and a structural link to OEIS sequence A047749.
• Paul Johnson’s work revealed that the odd-part sequence is related to the even-part sequence, enabling a fast algorithm to compute the sequence from the even-part generating function.
• For $k=2$, the generating function is $-\frac{x^4 - x^3 - x^2 + x + 1}{x^5 - x^4 - 2x^3 + 3x^2 + x - 1}$, with first 20 coefficients matching the odd-part sequence.
• For $k=3$, the generating function is $-\frac{x^9 + x^8 - 4x^7 - 6x^6 + 8x^5 + 9x^4 - 5x^3 - 5x^2 + x + 1}{(x^9 + 2x^8 - 3x^7 - 9x^6 + 3x^5 + 14x^4 - x^3 - 7x^2 + 1)(x - 1)}$, yielding the first 25 coefficients of the sequence.
• The authors made a $200 donation to the OEIS in honor of Paul Johnson for his discovery, which resolved the core challenge of computing the sequence efficiently.