[Paper Review] Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of a (2n + 1, 2n + 3)-core partition with Distinct Parts

Anthony Zaleski, Doron Zeilberger|arXiv (Cornell University)|Nov 17, 2016

Advanced Combinatorial Mathematics7 references4 citations

TL;DR

This paper provides explicit polynomial expressions for the expectation, variance, and higher moments (up to the seventh) of the size of $(2n+1, 2n+3)$-core partitions with distinct parts, using symbolic computation and experimental mathematics techniques. It confirms that these moments are polynomials in $n$, and reveals that the limiting distribution is not normal, with specific scaled limits for skewness and kurtosis derived from the moments.

ABSTRACT

Inspired by Armin Straub's conjecture (arXiv:1601.07161) about the number and maximal size of (2n+1, 2n+3)-core partitions with distinct parts, we develop relatively efficient, symbolic-computational algorithms, based on non-linear functional recurrences, to generate the generating functions, according to size, of the set of such partitions. By computing these polynomials for n=1,...21, we are able to rigorously derive explicit expressions for the expectation, variance, and third through seventh moments of the random variable "size of a (2n+1, 2n+3)-core partition with distinct parts." In particular, we find that this random variable is not asymptotically normal as n goes to infinity.

Motivation & Objective

• To derive explicit polynomial expressions for the expectation, variance, and higher moments of the size of $(2n+1, 2n+3)$-core partitions with distinct parts.
• To demonstrate that these moments are polynomials in $n$, in contrast to the Fibonacci-based expressions seen in the $(s,s+1)$-core case.
• To provide a faster, rigorizable alternative to prior complex combinatorial proofs of conjectures by Armin Straub on the number and maximal size of such partitions.
• To compute the first 21 Straub polynomials $S_n(q)$, the generating functions for these partitions, using non-linear functional recurrences.
• To investigate the limiting distribution of the size random variable and compute its scaled moments, revealing non-normality.

Proposed method

• The authors use symbolic computation via Maple packages `Armin.txt` and `core.txt` to generate the first 21 Straub polynomials $S_n(q)$, which encode the generating function for $(2n+1, 2n+3)$-core partitions with distinct parts.
• They apply the umbral operator $(q\frac{d}{dq})^k$ to $S_n(q)$, evaluate at $q=1$, and normalize by $4^n$ to extract raw moments.
• The moments are then fitted to polynomial forms in $n$ using numerical sequences derived from $S_n(q)$, with degrees confirmed up to $n=21$.
• The method relies on experimental mathematics: empirical fitting of sequences is rigorized a posteriori by verifying polynomial identities over sufficiently many values.
• Non-linear functional recurrences for $S_n(q)$ are used to compute the polynomials efficiently, avoiding direct enumeration.
• The scaled limits of higher moments (skewness, kurtosis) are computed from the polynomial expressions of the standardized moments.

Experimental results

Research questions

• RQ1 What are the explicit polynomial expressions for the expectation, variance, and higher moments of the size of $(2n+1, 2n+3)$-core partitions with distinct parts?
• RQ2 Is the limiting distribution of the size of such partitions normal, or does it exhibit non-normal characteristics?
• RQ3 What are the exact values of the scaled limits of the third through seventh moments (skewness and kurtosis) of the size distribution?
• RQ4 Can the conjectured maximal size of such partitions be rigorously confirmed using symbolic computation and polynomial fitting?
• RQ5 Do the moments of the size distribution exhibit polynomial dependence on $n$, and if so, what are their degrees?

Key findings

• The expectation of the size of a $(2n+1, 2n+3)$-core partition with distinct parts is given by a cubic polynomial in $n$, explicitly computed as $\mu(n) = \frac{1}{24}(5n^3 + 15n^2 + 10n)$.
• The variance is a degree-6 polynomial in $n$, derived from the second moment and the square of the mean, with the expression confirmed up to $n=21$.
• The scaled limit of the coefficient of variation is $\sqrt{14010}/150 \approx 0.789092305$, indicating a non-normal limiting distribution.
• The scaled limit of the third moment (skewness) is $\frac{396793}{390815488} \cdot \sqrt{467 \cdot 7680} \approx 1.92278748$, confirming asymmetry in the limiting distribution.
• The scaled limit of the fourth moment (kurtosis) is $145309380/16792853 \approx 8.6530490$, significantly exceeding the kurtosis of the normal distribution.
• The maximal size of such partitions is given by the degree-4 polynomial $\frac{1}{24}(5n+11)n(n+2)(n+1)$, confirmed up to $n=21$ via the degree of $S_n(q)$.