8,658 research outputs found
Binomial Eulerian polynomials for colored permutations
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and
Williams on the face enumeration of generalized permutohedra. They are
-positive (in particular, palindromic and unimodal) polynomials which
can be interpreted as -polynomials of certain flag simplicial polytopes and
which admit interesting Schur -positive symmetric function
generalizations. This paper introduces analogues of these polynomials for
-colored permutations with similar properties and uncovers some new
instances of equivariant -positivity in geometric combinatorics.Comment: Final version; minor change
Some applications of Rees products of posets to equivariant gamma-positivity
The Rees product of partially ordered sets was introduced by Bj\"orner and
Welker. Using the theory of lexicographic shellability, Linusson, Shareshian
and Wachs proved formulas, of significance in the theory of gamma-positivity,
for the dimension of the homology of the Rees product of a graded poset
with a certain -analogue of the chain of the same length as . Equivariant
generalizations of these formulas are proven in this paper, when a group of
automorphisms acts on , and are applied to establish the Schur
gamma-positivity of certain symmetric functions arising in algebraic and
geometric combinatorics.Comment: Final version, with a section on type B Coxeter complexes added; to
appear in Algebraic Combinatoric
Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all
roots of its characteristic polynomial have the same real part. This property
was conjectured by Postnikov and Stanley for certain families of arrangements
which are defined for any irreducible root system and was proved for the root
system . The proof is based on an explicit formula for the
characteristic polynomial, which is of independent combinatorial significance.
Here our previous derivation of this formula is simplified and extended to
similar formulae for all but the exceptional root systems. The conjecture
follows in these cases
Asymptotic Behavior of Strategies in the Repeated Prisoner's Dilemma Game in the Presence of Errors
We examine the asymptotic behavior of a finite, but error-prone population, whose agents can choose one of ALLD (always defect), ALLC (always cooperate), or Pavlov (repeats the previous action if the opponent cooperated and changes action otherwise) to play the repeated Prisoner's Dilemma. A novelty of the study is that it allows for three types of errors that affect agents' strategies in distinct ways: (a) implementation errors, (b) perception errors of one's own action, and (c) perception errors of the opponent's action. We also derive numerical results based on the payoff matrix used in the tournaments of Axelrod (1984). Strategies' payoffs are monitored as the likelihood of committing errors increases from zero to one, which enables us to provide a taxonomy of best response strategies. We find that for some range of error levels, a unique best response (i.e. a dominant strategy) exists. In all other, the population composition can vary based on the proportion of each strategist's type and/or the payoffs of the matrix. Overall, our results indicate that the emergence of cooperation is considerably weak at most error levels
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