Permutation Involution Count Calculator

Use a(0) = a(1) = 1 and split on whether the newest element is fixed or paired with one of n-1 others, giving a(n)=a(n-1)+(n-1)a(n-2). The recurrence accounts for both permitted cycle lengths without enumerating permutations. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

An involution may contain fixed points; it is not the same set as derangements or unrestricted permutations. Use the previous two values and multiply the pairing branch by n minus one. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

n is capped at 20 to keep factorial-like counting exact and browser work predictable. Involution counts eventually grow quickly despite the compact recurrence. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.