Information on Involutions
An involution, on a set S is a permutation
(i.e., a bijection), p, on S for which
p(p(s)) = s for all s in S.
In other words, they are the self-inverse permutations.
In terms of the cycle-structure of the permutation,
every cycle is of length 1 or 2.
Involutions are characterized by the property that their
permutation matrices are symmetric.
The number of involutions on an n-set, call it
I(n), satisfies the recurrence relation
I(n) = I(n-1) + (n-1) I(n-2).
The Schensted correspondence can be used to show a one-to-one
correspondence between involutions and standard
The number of involutions for n = 1,2,...,15, is
1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152,
568504, 2390480, 10349536.
This is sequence
Neil J. Sloane's
of integer sequences.
The algorithm used is from the paper:
U. Taylor and F. Ruskey,
Efficient Generation of Restricted Classes of Permutations,
The wizard of COS.
(Please note that the suffix XXXX must be removed from the preceeding
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©Frank Ruskey, 1995-2003.