Let 
(where 
)
be the disjoint union of two finite components
and 
.
Let 
and 
be two involutions on 
, each of whose fixed points lie in 
. Let 
(respectively, 
) denote the fixed point set of 
(respectively, 
). Stipulate that 
and 
, and similarly 
and 
(i.e., outside the fixed point sets), both
and 
map each component into the other. Then either a cycle of the permutation 
contains no fixed points of either 
or 
, or it contains exactly one
element of 
and one of 
.
Garsia-Milne Involution Principle
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References
Andrews, G. E. "q-Series and Schur's Theorem" and "Bressoud's Proof of Schur's Theorem." §6.2-6.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 53-58, 1986.Referenced on Wolfram|Alpha
Garsia-Milne Involution PrincipleCite this as:
Weisstein, Eric W. "Garsia-Milne Involution Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Garsia-MilneInvolutionPrinciple.html