A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$

We construct a statistic-swapping involution on the Cartesian product of the generalized symmetric group $S(k,n)$ with the symmetric group $S_{kn}$, which swaps the number of fixed points in the generalized symmetric group element with the number of $k$-cycles in the symmetric group element. This gives a combinatorial proof for a probabilistic observation: the distribution of fixed points on $S(k,n)$ matches the distribution of $k$-cycles on $S_{kn}$.

💡 Research Summary

The paper establishes a combinatorial involution that swaps two natural statistics on two different families of permutation‑like objects: the number of k‑cycles in a permutation of the symmetric group S_{kn} and the number of fixed points in an element of the generalized symmetric group S(k,n)=ℤ_k≀S_n. The authors begin by recalling that a fixed point of σ=(x,τ)∈S(k,n) is an index i for which the colour component x_i is zero and the underlying permutation τ fixes i. Their main result, Theorem 1.2, asserts that for every m the set of permutations in S_{kn} with exactly m k‑cycles and the set of elements in S(k,n) with exactly m fixed points have the same cardinality. Consequently, the distributions of these two statistics coincide under uniform sampling.

To prove this, the authors construct an auxiliary bijection
 f : S_{kn} → D_{k,n} × S(k,n)
where D_{k,n} consists of permutations of kn that are a product of n disjoint k‑cycles. The map f is defined in two parts. The first part f₁ applies Stanley’s fundamental bijection (which turns a permutation written in canonical cycle notation into a word by deleting parentheses) and then groups the resulting word into blocks of length k, re‑inserting parentheses to obtain a permutation δ∈D_{k,n}. The second part f₂ extracts from the same word a sequence τ of “block leaders” (the maximal element of each block) and, after applying the inverse of Stanley’s bijection, obtains a permutation τ∈S_n. Simultaneously, a colour vector x∈ℤ_k^n is defined: x_i records, modulo k, the distance from the block leader τ_i to the end of its original cycle in the input permutation π. The pair (x,τ) is an element of S(k,n). The authors prove a series of lemmas (2.1–2.4) showing that a block leader τ_i starts a k‑cycle in π if and only if the corresponding entry in (x,τ) is a fixed point, and therefore the number of k‑cycles of π equals the number of fixed points of f₂(π).

Having established that f preserves the statistic of interest, the involution ϕ on the Cartesian product S(k,n)×S_{kn} is defined by
 ϕ(σ′,π) = ( f₂(π) ,  f⁻¹( f₁(π) , σ′ ) ).
Because f is a bijection, ϕ is its own inverse. Moreover, by construction the k‑cycle count of π becomes the fixed‑point count of the new σ′, and vice versa. This directly yields the bijective proof of the equality of the two distributions claimed in Theorem 1.2.

The paper also details how to invert f. After obtaining (δ,x,τ) from a given π, one recovers the original word ˆπ by reordering the blocks of δ according to τ and then cyclically shifting each block by an amount s_i determined uniquely from the colour vector x. Lemma 3.2 guarantees the existence and uniqueness of these shifts. An explicit example walks through the entire process, confirming that the original permutation is reconstructed exactly.

In the concluding discussion the authors note that their construction not only provides a clean combinatorial explanation for the probabilistic observation that fixed points in S(k,n) and k‑cycles in S_{kn} share the same distribution, but also showcases a versatile technique: combining Stanley’s fundamental bijection with a block‑wise encoding of permutation structure. They suggest that similar involutions could be designed for other statistics (e.g., total colour sum, major index) and that the method may extend to other wreath products or to refined distributional results. Overall, the paper contributes a concrete, elementary bijection that bridges two seemingly unrelated permutation statistics, enriching the toolbox for algebraic combinatorics and probabilistic group theory.