97 research outputs found
The area above the Dyck path of a permutation
In this paper we study a mapping from permutations to Dyck paths. A Dyck path
gives rise to a (Young) diagram and we give relationships between statistics on
permutations and statistics on their corresponding diagrams. The distribution
of the size of this diagram is discussed and a generalisation given of a parity
result due to Simion and Schmidt. We propose a filling of the diagram which
determines the permutation uniquely. Diagram containment on a restricted class
of permutations is shown to be related to the strong Bruhat poset.Comment: 9 page
Restricted Dumont permutations, Dyck paths, and noncrossing partitions
We complete the enumeration of Dumont permutations of the second kind
avoiding a pattern of length 4 which is itself a Dumont permutation of the
second kind. We also consider some combinatorial statistics on Dumont
permutations avoiding certain patterns of length 3 and 4 and give a natural
bijection between 3142-avoiding Dumont permutations of the second kind and
noncrossing partitions that uses cycle decomposition, as well as bijections
between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths.
Finally, we enumerate Dumont permutations of the first kind simultaneously
avoiding certain pairs of 4-letter patterns and another pattern of arbitrary
length.Comment: 20 pages, 5 figure
Deformed Kazhdan-Lusztig elements and Macdonald polynomials
We introduce deformations of Kazhdan-Lusztig elements and specialised
nonsymmetric Macdonald polynomials, both of which form a distinguished basis of
the polynomial representation of a maximal parabolic subalgebra of the Hecke
algebra. We give explicit integral formula for these polynomials, and
explicitly describe the transition matrices between classes of polynomials. We
further develop a combinatorial interpretation of homogeneous evaluations using
an expansion in terms of Schubert polynomials in the deformation parameters.Comment: major revision, 29 pages, 22 eps figure
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Dyck tilings, increasing trees, descents, and inversions
Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon
tiles shaped like Dyck paths, in which tiles are no larger than the tiles they
cover. These tilings arise in the study of certain statistical physics models
and also Kazhdan--Lusztig polynomials. We give two bijections between
cover-inclusive Dyck tilings and linear extensions of tree posets. The first
bijection maps the statistic (area + tiles)/2 to inversions of the linear
extension, and the second bijection maps the "discrepancy" between the upper
and lower boundary of the tiling to descents of the linear extension.Comment: 24 pages, 9 figure
A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1
whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We
present a unifying perspective on ASMs and other combinatorial objects by
studying a certain tetrahedral poset and its subposets. We prove the order
ideals of these subposets are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self-complementary
plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux,
Catalan objects, tournaments, and totally symmetric plane partitions. We prove
product formulas counting these order ideals and give the rank generating
function of some of the corresponding lattices of order ideals. We also prove
an expansion of the tournament generating function as a sum over TSSCPPs. This
result is analogous to a result of Robbins and Rumsey expanding the tournament
generating function as a sum over alternating sign matrices.Comment: 24 pages, 23 figures, full published version of arXiv:0905.449
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