Discrete Morse theory and the consecutive pattern poset
Bruce E. Sagan
and Robert Willenbring
Department of Mathematics, Michigan State University, East Lansing, MI, 48824-1027, USA
DOI: 10.1007/s10801-012-0347-3
Abstract
We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrímsson's formula for the Möbius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Björner determined the Möbius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Björner's result. Since our proof parallels Willenbring's, we also consider the relationship between the two posets. In particular, we show that some of their intervals are isomorphic, and also that there is a sequence of posets interpolating between the two all of whom have essentially the same Möbius function.
Pages: 501–514
Keywords: consecutive pattern; Möbius function; discrete Morse theory; factor order; permutation patterns; posets
Full Text: PDF
References
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CrossRef Forman, R.: A discrete Morse theory for cell complexes. In: Geometry, Topology, \& Physics. Conf. Proc. Lecture Notes Geom. Topology, vol. IV, pp. 112-125. Int. Press, Cambridge (1995) Forman, R.: A user's guide to discrete Morse theory. Sémin. Lothar. Comb. 48, B48c (2002). 35 pp. (electronic) Simion, R., Schmidt, F.W.: Restricted permutations. Eur. J. Comb. $6(4)$, 383-406 (1985) Willenbring, R.: The Möbius function of generalized factor order.
arXiv:1108.3899
