39 research outputs found
Flag-symmetry of the poset of shuffles and a local action of the symmetric group
We show that the poset of shuffles introduced by Greene in 1988 is
flag-symmetric, and we describe a "local" permutation action of the symmetric
group on the maximal chains which is closely related to the flag symmetric
function of the poset. A key tool is provided by a new labeling of the maximal
chains of a poset of shuffles, which is also used to give bijective proofs of
enumerative properties originally obtained by Greene. In addition we define a
monoid of multiplicative functions on all posets of shuffles and describe this
monoid in terms of a new operation on power series in two variables.Comment: 34 pages, 6 figure
EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets
We show that a finite graded lattice of rank n is supersolvable if and only
if it has an EL-labeling where the labels along any maximal chain form a
permutation. We call such a labeling an S_n EL-labeling and we consider finite
graded posets of rank n with unique top and bottom elements that have an S_n
EL-labeling. We describe a type A 0-Hecke algebra action on the maximal chains
of such posets. This action is local and gives a representation of these Hecke
algebras whose character has characteristic that is closely related to
Ehrenborg's flag quasi-symmetric function. We ask what other classes of posets
have such an action and in particular we show that finite graded lattices of
rank n have such an action if and only if they have an S_n EL-labeling.Comment: 18 pages, 8 figures. Added JCTA reference and included some minor
corrections suggested by refere
Decomposition and enumeration in partially ordered sets
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 123-126).by Patricia Hersh.Ph.D
Enumeration of the distinct shuffles of permutations
A shuffle of two words is a word obtained by concatenating the two original words in either order and then sliding any letters from the second word back past letters of the first word, in such a way that the letters of each original word remain spelled out in their original relative order. Examples of shuffles of the words and are, for instance, and . In this paper, we enumerate the distinct shuffles of two permutations of any two lengths, where the permutations are written as words in the letters and , respectively
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Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
This memoir constitutes the author's PhD thesis at Cornell University. It
serves both as an expository work and as a description of new research. At the
heart of the memoir, we introduce and study a poset for each
finite Coxeter group and for each positive integer . When , our
definition coincides with the generalized noncrossing partitions introduced by
Brady-Watt and Bessis. When is the symmetric group, we obtain the poset of
classical -divisible noncrossing partitions, first studied by Edelman.
Along the way, we include a comprehensive introduction to related background
material. Before defining our generalization , we develop from
scratch the theory of algebraic noncrossing partitions . This involves
studying a finite Coxeter group with respect to its generating set of
{\em all} reflections, instead of the usual Coxeter generating set . This is
the first time that this material has appeared in one place.
Finally, it turns out that our poset shares many enumerative
features in common with the ``generalized nonnesting partitions'' of
Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In
particular, there is a generalized ``Fuss-Catalan number'', with a nice closed
formula in terms of the invariant degrees of , that plays an important role
in each case. We give a basic introduction to these topics, and we describe
several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical
Society. Many small improvements in exposition, especially in Sections 2.2,
4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor