[Paper Review] Proof of a conjecture of Stanley about Stern's array

David E Speyer|arXiv (Cornell University)|Jan 18, 2019

Algorithms and Data Compression2 references4 citations

TL;DR

This paper proves a conjecture by Richard Stanley that the sum $ s_n^r = igsum_k iglangle n \atop k \big angle^r $, where Stern's array entries are raised to the $ r $-th power and summed over the $ n $-th row, satisfies a homogeneous linear recurrence of length $ r/3 + O(1) $, significantly shorter than the previously known bound of $ r/2 + O(1) $. The proof uses generating functions, linear algebra over $ ext{SL}(2,bZ) $, and representation theory of the modular group to analyze the recurrence structure via eigenvalue bounds on a linear operator $ ilde{ ho}^* $.

ABSTRACT

Stanley, building on work of Stern, defined an array of numbers by the recurrence $s(n, 2k) = s(n-1, k)$, $s(n, 2k+1) = s(n-1, k) + s(n-1, k+1)$. Stanley showed that, for each positive integer $r$, the sequence $s_n^r:= \sum_k s(n,k)^r$ obeys a homogeneous linear recurrence in $n$ of length $r/2+O(1)$. Numerical evidence, however, suggested that $s_n^r$ obeys shorter recurrences, of length $r/3+O(1)$. We prove Stanley's conjecture.

Motivation & Objective

• To resolve Stanley's conjecture that the $ r $-th power sum $ s_n^r $ of entries in Stern's array satisfies a linear recurrence of length $ r/3 + O(1) $, rather than the previously known $ r/2 + O(1) $.
• To understand the algebraic structure underlying the recurrence relations of power sums in Stern's array using linear operators and group representations.
• To establish sharp bounds on the multiplicity of eigenvalues of a linear operator $ ilde{ ho}^* $ acting on homogeneous polynomials, which governs the recurrence length.
• To analyze the symmetric and antisymmetric subspaces of the polynomial space to refine recurrence bounds for even and odd $ r $.
• To unify the analysis of $ s_n^r $ for both even and odd $ r $, using characteristic polynomials and block-diagonalization of the recurrence operator.

Proposed method

• Define the sum $ s_n^r = igsum_k iglangle n \atop k \big angle^r $, where $ iglangle n \atop k \big angle $ are entries of Stern's array generated by a recursive insertion rule.
• Introduce a linear operator $ ilde{ ho}^* $ acting on homogeneous polynomials of degree $ r $, with $ ilde{ ho}^*(f)(x,y) = f(x+y, x) $, and analyze its action on the space of polynomials.
• Use the action of $ ext{SL}(2,bZ) $ via $ ilde{ ho}^* $, $ au^* $, and $ ho^* $ to model the recurrence structure of $ s_n^r $, reducing the problem to eigenvalue analysis.
• Compute the dimension of invariant subspaces $ X $, $ Y^+ $, and $ Y^- $, and use inclusion-exclusion to bound the dimension of their intersections, yielding lower bounds on the multiplicity of eigenvalues $ ho = ilde{ ho}^* $.
• Analyze the symmetric and antisymmetric versions $ ilde{ ho}_{ ext{sym}}^* $, identifying quotient spaces that reduce dimension by half and refine recurrence bounds.
• Use block-diagonalization and characteristic polynomials of the recurrence operator $ ilde{ ho}_{ ext{sym}}^* $ to derive recurrence lengths, showing that $ s_n^r $ satisfies a recurrence of length $ r/3 + O(1) $.

Experimental results

Research questions

• RQ1 Does the sum $ s_n^r = igsum_k iglangle n \atop k \big angle^r $ of $ r $-th powers of entries in the $ n $-th row of Stern's array satisfy a linear recurrence of length $ r/3 + O(1) $, as numerically suggested?
• RQ2 What is the precise algebraic structure governing the recurrence length of $ s_n^r $, and how do eigenvalues of the associated linear operator $ ilde{ ho}^* $ determine this?
• RQ3 How do the symmetric and antisymmetric subspaces of the polynomial space affect the recurrence length, particularly for even and odd $ r $?
• RQ4 Can the recurrence length be bounded using representation theory of $ ext{SL}(2,bZ) $, specifically via the action of $ ilde{ ho}^* $ on homogeneous polynomials?
• RQ5 What is the minimal recurrence length satisfied by $ s_n^r $, and how does it compare to the previously known bound of $ r/2 + O(1) $? Is it tight?

Key findings

• The sum $ s_n^r = igsum_k iglangle n \atop k \big angle^r $ satisfies a homogeneous linear recurrence of length $ r/3 + O(1) $, confirming Stanley's conjecture.
• The recurrence length is bounded below by $ frac{r}{3} + [-1, frac{1}{3}, frac{5}{3}]_r $ for the general case, and $ frac{r}{6} + [-1, - frac{1}{3}, frac{1}{3}]_r $ for the symmetric case.
• The multiplicity of eigenvalues $ ho = ilde{ ho}^* $ is bounded via dimension estimates on intersections of subspaces $ X igcap Y^+ $ and $ X igcap Y^- $, yielding $ frac{r}{6} + [0, - frac{1}{3}, frac{4}{3}, -1, frac{2}{3}, frac{1}{3}]_r $ and similar for $ Y^- $.
• For odd $ r $, the recurrence length is bounded by $ frac{r}{3} + [4, frac{10}{3}, frac{8}{3}]_r $, derived from the characteristic polynomial of $ ilde{ ho}_{ ext{sym}}^* $.
• The symmetric version $ ilde{ ho}_{ ext{sym}}^* $ reduces the dimension by half, and the recurrence length for $ s_n^r $ is bounded by $ frac{r}{6} + [-1, - frac{1}{3}, frac{1}{3}]_r $, confirming the $ r/3 $ scaling.
• The recurrence for $ s_n^r $ is shown to be of the form $ s_n = igsum_{j=1}^ u a_j s_{n-j} + b + c(-1)^n $, with $ u o r/3 $ as $ r $ grows, confirming the asymptotic bound.