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Adding--swapping mappings for $k$-th power permutations in $S_n$

Recently, Luo [3] introduced the Adding–Swapping Mapping Method to provide an alternative and constructive proof of Stanley’s [4] conjecture on perfect square permutations in $S_n$ and asked whether the method extends to higher powers. In this paper we answer that question in a more limited but precise structural sense. For each fixed $k\ge 2$, we define the $k$-signature $R_k(w)$ recording the cycle-count vector modulo $\gcd(m,k)$ in each length $m$, and we prove a local residue transition law describing how the insertion map $D_i$ updates the signature once the cycle length of the insertion point is specified. We also prove explicitly that every $k$-th power permutation has zero $k$-signature, so the signature gives a necessary obstruction to being a $k$-th power. This yields a residue-based partition of $S_n$ that serves as an indexing scheme for insertion updates. We then show that for $k\ge 3$ the insertion family does not preserve the class of $k$-th powers, explaining why the square case is exceptional from the standpoint of Luo’s method. Finally, we include explicit small-$n$ data for $k=3,4$ and prove that the density of $k$-th powers in $S_n$ tends to $0$ as $n\to\infty$.

Combinatorial Press

Yomi Anifowoshe Chong Han

Ars Combinatoria

2026

Title: Adding--swapping mappings for $k$-th power permutations in $S_n$

Description:

In this paper we answer that question in a more limited but precise structural sense.

For each fixed $k\ge 2$, we define the $k$-signature $R_k(w)$ recording the cycle-count vector modulo $\gcd(m,k)$ in each length $m$, and we prove a local residue transition law describing how the insertion map $D_i$ updates the signature once the cycle length of the insertion point is specified.

We also prove explicitly that every $k$-th power permutation has zero $k$-signature, so the signature gives a necessary obstruction to being a $k$-th power.

This yields a residue-based partition of $S_n$ that serves as an indexing scheme for insertion updates.

We then show that for $k\ge 3$ the insertion family does not preserve the class of $k$-th powers, explaining why the square case is exceptional from the standpoint of Luo’s method.

Finally, we include explicit small-$n$ data for $k=3,4$ and prove that the density of $k$-th powers in $S_n$ tends to $0$ as $n\to\infty$.

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Adding--swapping mappings for \(k\)-th power permutations in \(S_n\)

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