Is it possible for a permutation on n symbols to contain exactly n

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.

Let $m,n\in\mathbb N$ with $n\geq m$. For permutations $\sigma\in S_n$ and $\tau\in S_m$, let $X_\tau(\sigma)$ count the index sets $i_1<\cdots<i_m$ such that for every $p,q\in[m]$, $\sigma(i_p)<\sigma(i_q)$ if and only if $\tau(p)<\tau(q)$. Call $\sigma$ perfectly $m$-symmetric if \[X_\tau(\sigma)=\frac1{m!}\binom nm=\frac{n!}{(m!)^2(n-m)!}\quad\text{for every }\tau\in S_m.\] The restriction $n\geq m$ excludes the empty-pattern-count examples.

Statement recorded in the supplied source.

For each integer $m>3$, does there exist an integer $n\geq m$ and a perfectly $m$-symmetric permutation in $S_n$? Are there infinitely many such permutations, equivalently such examples of unbounded order $n$, for each fixed $m>3$? The source reports existence for $m\leq3$. [S2]

Short English statement

For each pattern size above three, can all permutation patterns occur equally often in one permutation, and can this happen at arbitrarily large orders?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100034 (AMR-030-0034). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joshua N. Cooper and Andrew Petrarca, Symmetric and Asymptotically Symmetric Permutations (2008), section 1, exact expected pattern counts; section 2, definitions of nontrivial k-symmetry and Conjecture 1, pp. 1--2. https://arxiv.org/abs/0801.4181.