Arnold sequence as an f-transform

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge1$ and $1\le i\le n$, let $e_{n,i}$ count permutations $\sigma$ of $\{1,\ldots,n\}$ such that $\sigma(n)=i$ and $\sigma(j-1)<\sigma(j)$ for even $j$, while $\sigma(j-1)>\sigma(j)$ for odd $j$. In particular $e_{1,1}=1$; equivalently, for $n\ge2$, \[ e_{n,i}=\begin{cases}\sum_{j<i}e_{n-1,j},&n\text{ even},\\\sum_{j\ge i}e_{n-1,j},&n\text{ odd}.\end{cases} \] In these sums $1\le j\le n-1$. Write $v_2(m)$ for the exponent of $2$ in a positive integer $m$, with $v_2(0)=+\infty$. The $i$th diagonal of the valuation triangle consists of $v_2(e_{n,i})$ for $n\ge i$. Set \[ m_i=\min_{n\ge i}v_2(e_{n,i}),\qquad u_k=\sup\left(\{i\in\mathbb N:m_i<k\}\cup\{0\}\right)\quad(k\ge1), \] where the supremum is allowed to be $+\infty$. When finite this is the maximum used in Arnold's sequence. Thus a proposed equality with a finite integer also asserts the required finiteness. The recorded initial values of $u_k$ are $2,4,4,4,8,8,8,8,10,12,12,16,\ldots$.

Let $\mathbb Z_{\ge0}$ include zero. For a finite tuple $x=(x_1,\ldots,x_d)$ of nonnegative integers, define a tuple $f(x)$ of length $2d$ as follows. If every $x_i=x_d$, then \[ f(x)=(\underbrace{x_d,\ldots,x_d}_{d\text{ entries}},\underbrace{2x_d,\ldots,2x_d}_{d\text{ entries}}). \] Otherwise put $s=\max\{i\in\{1,\ldots,d-1\}:x_i\ne x_d\}$ and define \[ f(x)=(x_1,\ldots,x_d,x_1+x_d,\ldots,x_{s-1}+x_d,\underbrace{2x_d,\ldots,2x_d}_{d-s+1\text{ entries}}). \] The middle block is empty when $s=1$. Since $f$ retains the entire input as a prefix, its iterates determine a unique infinite sequence $X(x)$: the $k$th term is the $k$th entry of any $f^n(x)$ with $2^nd\ge k$. This sequence is the $f$-transform of $x$.

Statement recorded in the supplied source.

Is $u_k=X_k((2,4,4,4))$ for every integer $k\ge1$? [S2]

Short English statement

Does repeatedly applying the explicitly defined tuple-doubling transformation to the seed $(2,4,4,4)$ produce exactly Arnold's valuation sequence?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9100003 (AMR-090-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Sanjay Ramassamy, Modular periodicity of the Euler numbers and a sequence by Arnold, Arnold Mathematical Journal (2018) §§3.1--3.2 and 4, Lemma 3 and Conjecture 3, pp. 3--6. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/18-79/index.html.