3-bit Walsh permutation; conjugacy class 2+2

The elements in this conjugacy class are self-inverse and leave half of the elements fixed (i.e. there are two fixed points for one transposition).

The cardinality of this conjugacy class is determined by Sloane'sA134057(n) = . For n = 3 that is = 21, the number of 2-subsets of a 7-set.

The equations show a bijection between the permutations and the 2-subsets of .   ( is the bitwise XOR.)
are the positive fixed points.
is the XOR of the elements in any of the two cycles. The other two fixed points are one of the 21 subsets.
E.g. 134 has the cycles (2, 3) and (6, 7). 2⊕3 = 6⊕7 = 1, which is one of the fixed points. The other two are {4, 5}, so this is the subset.

The pattern of fixed and moved points corresponds to the rows 1..7 of a Walsh matrix. Permutations with the same fixed points are in the same column.

1 2 4 3 5 6 7






Walsh matrix
0 1 2 3 4 5 6 7
0
1 134 125 135
2 324 126 326
3 214 127 217
4 524 164 564
5 174 421 471
6 724 142 742
7 654 623 153

 
 

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Above Sloane'sA134057(n) has been expressed as a binomial, but it can also be expressed as a product .     (3 · 7 for n = 3)
That is the number of 0s in positive rows and columns of a binary Walsh matrix.
In the Walsh matrix shown above the positive rows correspond to , and the positive columns to the patterns of fixed points.