OEIS A000246: Permutations with only odd-length cycles

We explore A000246—the count of permutations of n elements whose cycle decomposition uses only odd-length cycles. Key ideas: every counted permutation is even (each odd-length cycle is an even permutation, and the product of evens is even); the remarkable equivalence with ballot permutations; the neat closed forms a(2m) = ((2m−1)!!)^2 and a(2m+1) = (2m+1)!!(2m−1)!!; and how these two very different descriptions count the same objects. We’ll walk through small n (1→1, 2→1, 3→3, 4→9, 5→45, 6→225), explain the double-factorial formulas, and touch on the recurrence and the broader connections that make this OEIS entry a beautiful bridge between cycle structure and ballot sequences.

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