Observably, the number of primitive sorting networks on $n$ elements (or the number of commutation classes of reduced words of the longest element of $S\_n$) is even for $3\leq n\leq 15$. These are all the values for which it has been computed so far. Is there an abstract way to see that it is always even?


Sequence is at <http://oeis.org/A006245>



Is the number of commutation classes of reduced words of the longest element of $S_n$ even for $n\geq 3$?
On the linked OEIS entry I find the following:



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> Also the number of mappings X:{{1..n} choose 3}->{+,-} such that for any four indices a < b < c < d, the sequence X(a,b,c), X(a,b,d), X(a,c,d), X(b,c,d) changes its sign at most once (see Felsner-Weil and Balko-Fulek-Kynčl reference). - Manfred Scheucher, Oct 20 2019
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There is an obvious fixed-point free involution on the set of such maps given by switching "+" and "-", assuming that $n \geq 3$ so that $\binom n 3 > 0$.


