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 ).

The cardinality of this conjugacy class is determined by (n) = $$\binom{2^n-1}{2}$$. For n = 3 that is $$\binom{7}{2}$$ = 21, the number of 2-subsets of a 7-set.

The equations $$a \oplus b ~=~ c$$ show a bijection between the permutations and the 2-subsets of $$\{1..7\}$$. ($$\oplus$$ is the .) $$\{a, b, c\}$$ are the positive fixed points. $$c$$ is the XOR of the elements in any of the two cycles. The other two fixed points $$\{a, b\}$$ 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. Permutations with the same fixed points are in the same column.

Above (n) has been expressed as a binomial, but it can also be expressed as a product $$(2^{n-1} -1) \cdot (2^n - 1)$$. (3 · 7 for n = 3) That is the number of 0s in positive rows and columns of a binary. In the Walsh matrix shown above the positive rows correspond to $$a \oplus b$$, and the positive columns to the patterns of fixed points.