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 Principle## Cite this as:

Weisstein, Eric W. "Garsia-Milne Involution Principle." From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Garsia-MilneInvolutionPrinciple.html