Are There n!/2 Even/Odd Permutation Matrices for nxn?

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Discussion Overview

The discussion revolves around the number of even and odd permutation matrices for an nxn matrix, specifically questioning whether this number is n!/2 for n greater than 3. Participants seek to understand the derivation of this result.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asks for a proof that the number of even and odd permutation matrices is n!/2 for n greater than 3.
  • Another participant states that there are n! permutation matrices in total and mentions a bijective function between odd and even permutations, suggesting that this leads to n!/2 for both types.
  • A repeated point emphasizes the total number of permutation matrices and the bijective relationship, indicating a method to derive the n!/2 result.
  • A participant seeks clarification on how to demonstrate the bijectiveness of the relationship between odd and even permutations.
  • Another participant suggests constructing the bijection by associating each even permutation with a unique odd permutation, providing an example to illustrate this point.

Areas of Agreement / Disagreement

Participants generally agree on the total number of permutation matrices being n! and the existence of a bijective relationship between odd and even permutations, but the details of the proof and the construction of the bijection remain under discussion.

Contextual Notes

Some assumptions about the nature of permutations and the definitions of odd and even permutations are not explicitly stated, which may affect the clarity of the arguments presented.

Terrell
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is there a proof that the number of even/odd permutation matrices of any nxn, where n is greater than 3, is n!/2? basically, i want to understand the derivation of n!/2. thank you!
 
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It is easy to show that (a) there are in total n! permutation matrices and (b) there is a bijective function between odd and even permutations (e.g. swap two images). Combine both and you get n!/2 odd and n!/2 even permutations.
 
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mfb said:
It is easy to show that (a) there are in total n! permutation matrices and (b) there is a bijective function between odd and even permutations (e.g. swap two images). Combine both and you get n!/2 odd and n!/2 even permutations.
how can i show the bijectiveness? sorry if my question is stupid
 
By constructing it. For each even permutation, find exactly one unique odd permutation, or vice versa. I gave an example how to do that.
 
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