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Annals of Combinatorics
2 (1998) 1-6
Orbifold Euler Characteristics and the Number of Commuting *m*-tuples in the Symmetric Groups
Jim Bryan
and Jason Fulman
Department of Mathematics,University of California,
Berkeley, CA 94720, USA
jbryan@math.berkeley.edu
Department of Mathematics,
Dartmouth College,
Hanover, NH 03755, USA
jason.e.fulman@dartmouth.edu
Received November 4, 1997
**AMS Subject Classification**:
55N20, 20B30
**Abstract.**
Generating functions for the number of commuting *m*-tuples in the
symmetric groups are obtained. We define a natural sequence of
"orbifold Euler characteristics" for a finite group *G* acting on a manifold *X*.
Our definition generalizes the ordinary Euler characteristic of *X/G*
and the string-theoretic orbifold Euler characteristic. Our formulae for
commuting *m*-tuples underlie formulae that generalize the results
of Macdonald and Hirzebruch-Höfer concerning the ordinary
and string-theoretic Euler characteristics of symmetric products.
**Keywords**: Euler characteristic, orbifold,
cohomology, symmetric groups
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