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The Enumeration of Maximally Clustered Permutations
Hugh Denoncourt1 and Brant C. Jones2
1Department of Mathematics, Box 395, Boulder, Colorado 80309-0395, USA
hugh.denoncourt@colorado.edu
2Department of Mathematics, One Shields Avenue, University of California, Davis, CA 95616, USA
brant@math.ucdavis.edu
Annals of Combinatorics 14 (1) pp.65-84 Springer, 2010
AMS Subject Classification: 05A15, 05E15, 20F55
Abstract:
The maximally clustered permutations are characterized by avoiding the classical permutation patterns {3421, 4312, 4321}. This class contains the freely braided permutations and the fully commutative permutations. In this work, we show that the generating functions for certain fully commutative pattern classes can be transformed to give generating functions for the corresponding freely braided and maximally clustered pattern classes. Moreover, this transformation of generating functions is rational. As a result, we obtain enumerative formulas for the pattern classes mentioned above as well as the corresponding hexagonavoiding pattern classes where the hexagon-avoiding permutations are characterized by avoiding {46718235, 46781235, 56718234, 56781234}.
Keywords: pattern avoidance, 2-sided weak Bruhat order, 321-hexagon, freely braided, maximally clustered

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