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 $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. 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 $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-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 $(P,\omega)$-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

The birational Lalanne-Kreweras involution

The Lalanne-Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the Lalanne-Kreweras involution. Actually, we show that the Lalanne-Kreweras involution is a special case of a more general operator, called rowvacuation, which acts on the antichains of any graded poset. Rowvacuation, like the closely related and more studied rowmotion operator, is a composition of toggles. We obtain the piecewise-linear and birational lifts of the Lalanne-Kreweras involution by using the piecewise-linear and birational toggles of Einstein and Propp. We show that the symmetry properties of the Lalanne-Kreweras involution extend to these piecewise-linear and birational lifts.Comment: 44 pages, 11 figures; v2: forthcoming, Algebraic Combinatoric

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 $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization $NC^{(k)}(W)$, we develop from scratch the theory of algebraic noncrossing partitions $NC(W)$. This involves studying a finite Coxeter group $W$ with respect to its generating set $T$ of {\em all} reflections, instead of the usual Coxeter generating set $S$. This is the first time that this material has appeared in one place. Finally, it turns out that our poset $NC^{(k)}(W)$ 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 $W$, 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