243 research outputs found
Some remarks on sign-balanced and maj-balanced posets
Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if
exactly half the linear extensions of P (regarded as permutations of 1,2,...,n)
are even permutations, i.e., have an even number of inversions. This concept
first arose in the work of Frank Ruskey, who was interested in the efficient
generation of all linear extensions of P. We survey a number of techniques for
showing that posets are sign-balanced, and more generally, computing their
"imbalance." There are close connections with domino tilings and, for certain
posets, a "domino generalization" of Schur functions due to Carre and Leclerc.
We also say that P is maj-balanced if exactly half the linear extensions of P
have even major index. We discuss some similarities and some differences
between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and
Conjecture 3.6 has been adde
Lower Bounds for Real Solutions to Sparse Polynomial Systems
We show how to construct sparse polynomial systems that have non-trivial
lower bounds on their numbers of real solutions. These are unmixed systems
associated to certain polytopes. For the order polytope of a poset P this lower
bound is the sign-imbalance of P and it holds if all maximal chains of P have
length of the same parity. This theory also gives lower bounds in the real
Schubert calculus through sagbi degeneration of the Grassmannian to a toric
variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision
A new family of posets generalizing the weak order on some Coxeter groups
We construct a poset from a simple acyclic digraph together with a valuation
on its vertices, and we compute the values of its M\"obius function. We show
that the weak order on Coxeter groups of type A, B, affine A, and the flag weak
order on the wreath product introduced by Adin,
Brenti and Roichman, are special instances of our construction. We conclude by
associating a quasi-symmetric function to each element of these posets. In the
and cases, this function coincides respectively with the
classical Stanley symmetric function, and with Lam's affine generalization
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
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