Let n be the alphabet length, let s_r denote the size of the sphere of radius r, and let binomnr be the binomial coefficient. The quantities fracs_rbinomnr are defined for 1 leqslant r leqslant n. Monotonicity conjecture. The sequence (fracs_rbinomnr)^1/r is decreasing for 1 leqslant r leqslant n. This would extend the endpoint bounds, which are attained at r=1 and r=n, and give a stronger monotonicity principle for the sphere sizes of mixed codes with finite alphabets.

Monotonicity conjecture for normalized mixed-code sphere sizes

About 4 years old · traced to 2022

Let $n$ be the alphabet length, let $s_r$ denote the size of the sphere of radius $r$ , and let $\binom{n}{r}$ be the binomial coefficient. The quantities $\frac{s_r}{\binom{n}{r}}$ are defined for $1\leqslant r\leqslant n$ . Monotonicity conjecture. The sequence

\left(\frac{s_r}{\binom{n}{r}}\right)^{1/r}

is decreasing for $1\leqslant r\leqslant n$ . This would extend the endpoint bounds, which are attained at $r=1$ and $r=n$ , and give a stronger monotonicity principle for the sphere sizes of mixed codes with finite alphabets.

References

Primary source

Yonatan Yehezkeally, Haider Al Kim, Sven Puchinger and Antonia Wachter-Zeh, “Bounds on Mixed Codes with Finite Alphabets”, arXiv:2212.09314 (2022).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written complete proof claims to settle the conjecture for every alphabet choice, but the claim has not been independently verified.

Yonatan Yehezkeally, Haider Al Kim, Sven Puchinger, and Antonia Wachter-Zeh stated the conjecture in 2022. It asks whether the normalized sphere sizes $\left(s_r/\binom{n}{r}\right)^{1/r}$ decrease with $r$ ; their paper explicitly labels this Conjecture 5.

Known results

Theorem 3 proves that $\frac{(r+1)s_{r+1}}{(n-r)s_r}$ is nonincreasing in $r$ .
Theorem 4 proves endpoint bounds for $s_r/\binom{n}{r}$ , with equality at $r=1$ and $r=n$ .
The paper gives no proof or counterexample for the conjectured root monotonicity.

Posted attempt

A reader-written argument claims a complete proof for all finite alphabet sizes, deriving the desired inequalities from Theorem 3. This attempt has not been independently verified.

Current status (as of August 2026): The conjecture has a complete-proof claim but no independent verification; the endpoint bounds and related ratio monotonicity are settled, while the conjecture itself remains unconfirmed.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

Complete proof. Let $q_1,\ldots,q_n\ge2$ , put $a_i=q_i-1>0$ , and let $s_r$ be the mixed-code sphere size. Then

s_r=\sum_{\substack{I\subseteq\{1,\ldots,n\}\\|I|=r}} \prod_{i\in I}a_i=e_r(a_1,\ldots,a_n).

Define

E_r=\frac{s_r}{\binom nr}\quad(0\le r\le n), \qquad E_0=1.

Theorem 3 of the source proves that

d_r=\frac{(r+1)s_{r+1}}{(n-r)s_r} =\frac{E_{r+1}}{E_r}, \qquad 0\le r<n,

is nonincreasing as a function of $r$ .

For $1\le r<n$ , it follows that

E_r=\prod_{j=0}^{r-1}d_j\ge d_r^{\,r},

because every factor $d_j$ with $j<r$ is at least $d_r$ . Therefore

E_{r+1}=E_rd_r\le E_r^{\,1+1/r},

and hence

\boxed{\left(\frac{s_{r+1}}{\binom n{r+1}}\right)^{1/(r+1)} \le \left(\frac{s_r}{\binom nr}\right)^{1/r}} \qquad(1\le r<n).

This is exactly the conjectured monotonicity for all alphabet sizes. Equivalently, it is Maclaurin's inequality for the positive numbers $q_i-1$ . Equality occurs precisely in the homogeneous case $q_1=\cdots=q_n$ .