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

From papers

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.

Progress summary

Open

The conjecture remains open: known work proves endpoint bounds and a related monotonicity statement, but no proof or counterexample has been publicly recorded.

The conjecture asks whether the normalized sphere sizes $\left(s_r/\binom{n}{r}\right)^{1/r}$ decrease with $r$ . It is stated explicitly in the paper Bounds on Mixed Codes with Finite Alphabets.

Known results

Theorem 3 proves that $\frac{(r+1)s_{r+1}}{(n-r)s_r}$ is decreasing in $r$ .
Theorem 4 gives endpoint bounds for the normalized sphere sizes, with equality at $r=1$ and $r=n$ .
The same source labels the requested monotonicity as Conjecture 5 and provides no proof or counterexample.

Current status (as of August 2026): The conjecture remains unsettled; the related ratio monotonicity and endpoint bounds are proved, but the full sequence monotonicity has no publicly verified proof or counterexample.

Sources

Sources & referencesView supporting material

Primary source

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

1 week ago

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 $q_1=\cdots=q_n$ .

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Shivam Patel · 1 day ago