Partially ordered patterns and compositions
Abstract
A partially ordered (generalized) pattern (POP) is a generalized pattern some of whose letters are incomparable, an extension of generalized permutation patterns introduced by Babson and Steingrimsson. POPs were introduced in the symmetric group by Kitaev [Partially ordered generalized patterns, Discrete Math. 298 (2005), 212229; Introduction to partially ordered patterns, Discrete Appl. Math., to appear], and studied in the set of $k$ary words by Kitaev and Mansour [Partially ordered generalized patterns and $k$ary words, Annals of Combinatorics 7 (2003) 191200]. Moreover, Kitaev et al. [S. Kitaev, T. McAllister and K. Petersen, Enumerating segmented patterns in compositions and encoding with restricted permutations, preprint] introduced segmented POPs in compositions. In this paper, we study avoidance of POPs in compositions and generalize results for avoidance of POPs in permutations and words. Specifically, we obtain results for the generating functions for the number of compositions that avoid shuffle patterns and multipatterns. In addition, we give the generating function for the distribution of the maximum number of nonoverlapping occurrences of a segmented POP $\tau$ (that is allowed to have repeated letters) among the compositions of $n$ with $m$ parts in a given set, provided we know the generating function for the number of compositions of $n$ with $m$ parts in the given set that avoid $\tau$. This result is a $q$analogue of the main result in [S. Kitaev, T. Mansour, Partially ordered generalized patterns and $k$ary words, Annals of Combinatorics 7 (2003) 191200].
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2006
 arXiv:
 arXiv:math/0610030
 Bibcode:
 2006math.....10030H
 Keywords:

 Mathematics  Combinatorics;
 05A05;
 05A15
 EPrint:
 10 pages