Diagonalpermutohedra.tex

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\begin{document}

\frontmatter

\title{Cellular diagonals of permutahedra}

\thanks{
BDO was partially supported by the French ANR grants ALCOHOL (ANR-19-CE40-0006), CARPLO (ANR-20-CE40-0007), HighAGT (ANR-20-CE40-0016) and S3 (ANR-20-CE48-0010). 
GLA and KS were supported by the Australian Research Council Future Fellowship FT210100256. 
KS was supported by an Australian Government Research Training Program (RTP) Scholarship.
VP was partially supported by the French ANR grant CHARMS (ANR-19-CE40-0017) and by the French--Austrian project PAGCAP (ANR-21-CE48-0020 \& FWF I 5788).
}

\author[B. Delcroix-Oger]{B\'er\'enice Delcroix-Oger}
\address[B\'{e}r\'{e}nice Delcroix-Oger]{Institut Montpelli\'erain Alexander Grothendieck, Universit\'e de Montpellier, France}
\email{[e-mail]}
\urladdr{\url{https://oger.perso.math.cnrs.fr/}}

\author[G. Laplante-Anfossi]{Guillaume Laplante-Anfossi}
\address[Guillaume Laplante-Anfossi]{School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia}
\email{[e-mail]}
\urladdr{\url{https://guillaumelaplante-anfossi.github.io/}}

\author[V. Pilaud]{Vincent Pilaud}
\address[Vincent Pilaud]{CNRS \& LIX, \'Ecole Polytechnique, Palaiseau, France}
\email{[e-mail]}
\urladdr{\url{http://www.lix.polytechnique.fr/~pilaud/}}

\author[K. Stoeckl]{Kurt Stoeckl}
\address[Kurt Stoeckl]{School of Mathematics and Statistics, The University of Melbourne, Victoria, Australia}
\email{[e-mail]}
\urladdr{\url{https://kstoeckl.github.io/}}


%    \date is required; it is the date received by the editor.
\date{\today}

\subjclass[2020]{05A15; 05C05; 18M70; 52B11; 52C35}
%    Recognition of the 2010 edition of the Mathematics Subject
%    Classification requires a version of amsbook.cls from July 2009
%    or later.  If "2010" is not recognized, please upgrade.

\keywords{Diagonal, permutahedron, hyperplane arrangement}

%\dedicatory{Dedication text (use \\[2pt] for line break if necessary)}

\begin{abstract}
We provide a systematic enumerative and combinatorial study of geometric cellular diagonals on the permutahedra. 

In the first part of the paper, we study the combinatorics of certain hyperplane arrangements obtained as the union of $\ell$ generically translated copies of the classical braid arrangement.
Based on Zaslavsky's theory, we derive enumerative results on the faces of these arrangements involving combinatorial objects named partition forests and rainbow forests.
This yields in particular nice formulas for the number of regions and bounded regions in terms of exponentials of generating functions of Fuss-Catalan numbers.
By duality, the specialization of these results to the case~$\ell = 2$ gives the enumeration of any geometric diagonal of the permutahedron.

In the second part of the paper, we study diagonals which respect the operadic structure on the family of permutahedra.
We show that there are exactly two such diagonals, which are moreover isomorphic.
We describe their facets by a simple rule on paths in partition trees, and their vertices as pattern-avoiding pairs of permutations.
We show that one of these diagonals is a topological enhancement of the Sanbeblidze--Umble diagonal, and unravel a natural lattice structure on their sets of facets.

In the third part of the paper, we use the preceding results to show that there are precisely two isomorphic topological cellular operadic structures on the families of operahedra and multiplihedra, and exactly two infinity-isomorphic geometric universal tensor products of homotopy operads and A-infinity morphisms.

\end{abstract}

\maketitle

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\tableofcontents



%    Include unnumbered chapters (preface, acknowledgments, etc.) here.
\include{acknowledgements}
\include{intro}

\mainmatter
%    Include main chapters here.
\include{combinatorics}
\include{diag}
\include{higher}

%\appendix
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%\include{}

\backmatter
%    Bibliography styles amsplain or author-year (using natbib) are
%    also acceptable.
\bibliographystyle{amsalpha}
\bibliography{diagonalsPermutahedra}
\label{sec:biblio}

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\end{document}

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