CalConverters

Even/Odd Permutations Calculator

Split the n! permutations of n distinct items into even and odd classes — always exactly n!/2 each — with the transposition-parity reasoning and steps.

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How to use this calculator

Type the number of distinct items n. The calculator finds n!n!, the total number of permutations, and splits it exactly in half: n!/2n!/2 are even permutations and n!/2n!/2 are odd — always an equal split, for any n≥2n \ge 2.

Tip: The split is exact, not approximate — swapping any two entries in a permutation always flips its parity, which pairs up the even and odd classes one-to-one.

What is an even or odd permutation?

Any permutation — any reordering — of a list can be reached from the original order by a sequence of transpositions: swaps that exchange exactly two elements and leave the rest alone. A permutation is:

  • Even if it can be reached using an even number of swaps.
  • Odd if it takes an odd number of swaps.

This might look ambiguous at first — couldn't you reach the same permutation with a different number of swaps? You can use a different count, but never a different parity: every valid sequence of swaps that produces a given permutation has the same evenness or oddness. Parity is a genuine, fixed property of the permutation itself, not of any one way of building it.

Why the split is exactly 50/50

Take any permutation and swap two of its entries. That one extra swap flips its parity: even becomes odd, odd becomes even. Doing this with a fixed pair of positions (say, always swap the first two entries) creates a pairing — every even permutation is matched with exactly one odd permutation (its swapped version), and every odd permutation is matched with exactly one even permutation the same way. Because that pairing covers every permutation exactly once, the two classes must be the same size:

even permutations=odd permutations=n!2\text{even permutations} = \text{odd permutations} = \frac{n!}{2}

This holds for every n≥2n \ge 2, no matter how large — the ratio never drifts toward one class as nn grows.

The alternating group

The even permutations of nn items, together with the rule "combine two permutations by applying one after the other," form a structure called the alternating group, written AnA_n. It has n!/2n!/2 elements — exactly the each count this calculator returns. Alternating groups are a cornerstone of abstract algebra: A5A_5, with 60 elements, is the smallest example of a simple non-abelian group, a fact tied to why the general quintic equation has no algebraic (radical) solution formula.

Worked example — 4 items

  1. Total permutations: 4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 24.
  2. Every permutation is reached from the identity by some number of swaps; swapping any two entries always flips even ↔ odd.
  3. That swap pairs every even permutation with a unique odd one, so the 24 permutations split exactly in half.
  4. 24/2=1224 / 2 = 12 even and 1212 odd.

So of the 24 permutations of 4 items, 12 are even and 12 are odd — the 12 even ones are exactly the elements of A4A_4.

Frequently asked questions

What is an even or odd permutation?
Every permutation of a set can be produced from the identity order by a sequence of transpositions (swaps of exactly two elements). A permutation is even if it can be built with an even number of swaps, and odd if it takes an odd number. This parity is well-defined — no matter which particular sequence of swaps you choose to build a given permutation, the number of swaps always has the same parity.
Why does the split always come out exactly 50/50?
Swapping any two entries of a permutation flips its parity — an even permutation becomes odd and vice versa. That swap gives a one-to-one pairing between every even permutation and a unique odd permutation, so for any n≥2n \ge 2 there are exactly as many of one class as the other: n!/2n!/2 each.
What is 4! split into even and odd permutations?
4!=244! = 24 total permutations of 4 items, split into 24/2=1224/2 = 12 even and 1212 odd.
What is the alternating group?
The alternating group AnA_n is the set of all even permutations of nn items, together with the operation of composing permutations. It has n!/2n!/2 elements and is one of the most-studied objects in abstract algebra — for example, A5A_5 (60 even permutations of 5 items) is the smallest non-abelian simple group.
Why doesn't the split apply to n = 0 or n = 1?
With 0 or 1 items there is only one possible arrangement — the empty arrangement or the single item sitting alone — and it counts as even (zero swaps needed). There is no odd permutation to pair it with, so "half of 1" isn't a meaningful split. The 50/50 rule starts at n=2n = 2.
Is permutation parity the same as its sign?
Yes, they're two names for the same idea. The sign of a permutation is +1+1 for an even permutation and −1-1 for an odd one; sign shows up directly in the Leibniz formula for a determinant, where it decides whether each term is added or subtracted.
What's the largest n this calculator can compute?
Up to n = 5000, where n!/2 already has thousands of digits. Every result uses exact big-integer arithmetic, and results over 15 digits are also shown in scientific notation so they stay readable.

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