164 research outputs found
Counting Self-Dual Interval Orders
In this paper, we present a new method to derive formulas for the generating
functions of interval orders, counted with respect to their size, magnitude,
and number of minimal and maximal elements. Our method allows us not only to
generalize previous results on refined enumeration of general interval orders,
but also to enumerate self-dual interval orders with respect to analogous
statistics.
Using the newly derived generating function formulas, we are able to prove a
bijective relationship between self-dual interval orders and upper-triangular
matrices with no zero rows. Previously, a similar bijective relationship has
been established between general interval orders and upper-triangular matrices
with no zero rows and columns.Comment: 20 page
Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations
The combined work of Bousquet-M\'elou, Claesson, Dukes, Jel\'inek, Kitaev,
Kubitzke and Parviainen has resulted in non-trivial bijections among ascent
sequences, (2+2)-free posets, upper-triangular integer matrices, and
pattern-avoiding permutations. To probe the finer behavior of these bijections,
we study two types of restrictions on ascent sequences. These restrictions are
motivated by our results that their images under the bijections are natural and
combinatorially significant. In addition, for one restriction, we are able to
determine the effect of poset duality on the corresponding ascent sequences,
matrices and permutations, thereby answering a question of the first author and
Parviainen in this case. The second restriction should appeal to Catalaniacs.Comment: 24 pages, 4 figures. To appear in Journal of Combinatorial Theory,
Series
On naturally labelled posets and permutations avoiding 12-34
A partial order on is naturally labelled (NL) if
implies . We establish a bijection between {3, 2+2}-free NL posets and
12-34-avoiding permutations, determine functional equations satisfied by their
generating function, and use series analysis to investigate their asymptotic
growth. We also exhibit bijections between 3-free NL posets and various other
objects, and determine their generating function. The connection between our
results and a hierarchy of combinatorial objects related to interval orders is
described.Comment: 19 page
Ascent sequences and upper triangular matrices containing non-negative integers
This paper presents a bijection between ascent sequences and upper triangular
matrices whose non-negative entries are such that all rows and columns contain
at least one non-zero entry. We show the equivalence of several natural
statistics on these structures under this bijection and prove that some of
these statistics are equidistributed. Several special classes of matrices are
shown to have simple formulations in terms of ascent sequences. Binary matrices
are shown to correspond to ascent sequences with no two adjacent entries the
same. Bidiagonal matrices are shown to be related to order-consecutive set
partitions and a simple condition on the ascent sequences generate this class.Comment: 13 page
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