Standard paths in another composition poset

Bergeron, Bousquet-Melou and Dulucq enumerated paths in the Hasse diagram of the following poset: the underlying set is that of all compositions, and a composition \mu covers another composition \lambda if \mu can be obtained from \lambda by adding 1 to one of the parts of \lambda, or by inserting a part of size 1 into \lambda. We employ the methods they developed in order to study the same problem for the following poset: the underlying set is the same, but \mu covers \lambda if \mu can be obtained from \lambda by adding 1 to one of the parts of \lambda, or by inserting a part of size 1 at the left or at the right of \lambda. This poset is of interest because of its relation to non-commutative term orders.Comment: 9 page

Saturated chains in composition posets

We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The resulting poset was studied by the author in "A poset classifying non-commutative term orders", and then in "Standard paths in another composition poset" where some results about generating functions for standard paths in this poset was established. The latter article was inspired by the work of Bergeron, Bousquet-M{\'e}lou and Dulucq on "Standard paths in the composition poset", where they studied a poset where there are additional cover relations which allows the insertion of a part of size one anywhere in the composition. Finally, following a suggestion by Richard Stanley we study yet a third which is an extension of the previous two posets. This poset is related to quasi-symmetric functions. For these posets, we study generating functions for saturated chains of fixed width k. We also construct ``labeled'' non-commutative generating functions and their associated languages.Comment: 37 page

Reconstructing compositions

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition w of n\ge 3k+1 can be reconstructed from its set of k-deletions, i.e., the set of all compositions of n-k contained in w. As there are compositions of 3k with the same set of k-deletions, this result is best possible

On the dimension of downsets of integer partitions and compositions

We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension

Pattern Avoidance in Set Partitions

The study of patterns in permutations in a very active area of current research. Klazar defined and studied an analogous notion of pattern for set partitions. We continue this work, finding exact formulas for the number of set partitions which avoid certain specific patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us to conclude that the corresponding sequences are P-recursive. Finally, we define a second notion of pattern in a set partition, based on its restricted growth function. Related results are obtained for this new definition.Comment: 15 pages, see related papers at http://www.math.msu.edu/~saga