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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Discretized configurations and partial partitions


Authors: Aaron Abrams, David Gay and Valerie Hower
Journal: Proc. Amer. Math. Soc. 141 (2013), 1093-1104
MSC (2010): Primary 55R80, 05A18, 11B73
Published electronically: July 17, 2012
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Abstract: We show that the discretized configuration space of $ k$ points in the $ n$-simplex is homotopy equivalent to a wedge of spheres of dimension $ n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $ \{1,\dots ,n+1\}$ with exactly $ k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.


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Additional Information

Aaron Abrams
Affiliation: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Robinson Hall, Washington and Lee University, Lexington, Virginia 24450
Email: abrams.aaron@gmail.com

David Gay
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: d.gay@euclidlab.org

Valerie Hower
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Miami, Coral Gables, Florida 33146
Email: vhower@math.berkeley.edu, vhower@math.miami.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-10816-0
PII: S 0002-9939(2012)10816-0
Received by editor(s): September 15, 2010
Received by editor(s) in revised form: July 21, 2011
Published electronically: July 17, 2012
Communicated by: Ken Ono
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.