Smith normal form of the n-cycle class sum

We determine the Smith normal form of multiplication by the sum of all $n$-cycles on the center of $\mathbb{Z} S_n$. For every $n\geq1$, it has $p(n)-n$ zero entries, where $p(n)$ counts partitions of $n$, and nonzero factors $d_0=1$ and $d_r=a_r r!/\gcd(n,r)$ for $1\le r<n$, with $a_r=2$ when $r\equiv n\pmod2$ and $a_r=1$ otherwise. This confirms Stanley's conjectures for odd primes and twice odd primes and gives the $n$-cycle formula for every positive $n$. Goupil and Schaeffer gave a formula for long-cycle connection coefficients, and Biane gave an algebraic proof. We determine the two integral coefficient lattices needed to extract its Smith factors, then prove rowwise divisibility. Grinberg's exposition supplies class-sum and representation-theory background used here. Lean proves the image-lattice statement and divisor chain directly from group-ring definitions, including the character formula. The diagonal form and prime cases follow by written arguments.

Checked in Lean from the definitions; not yet reviewed by experts. Produced by AI models under the author's direction.