Reiner and Burgiel's recent paper provoked the questions at the beginning and end of the following: ************************************************************ Until August 10, 1998, I am on leave from UNC and am at the University of Pennsylvania Jim Stasheff jds@math.upenn.edu 146 Woodland Dr Lansdale PA 19446 (215)822-6707 Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 ---------- Forwarded message ---------- Date: Tue, 8 Jul 1997 11:33:31 -0500 (CDT) From: Victor Reiner <reiner@math.umn.edu> To: jds@math.upenn.edu Cc: burgiel@math.uic.edu Subject: Signed associahedra Dear Jim,
Only thing more I could ask is to have the tree descrition in addition to the triangulation description of the cells.
I think we can oblige. A signed dissection of the (n+2)-gon corresponds to a plane tree T having n+1 leaves in which one assigns +,-, or 0 to every "nook" between two branches of the tree (think of the +,-,0's as being like lint trapped between ones toes!). For example, a vertex having 5 children in the tree will have 4 nooks between its branches, and hence require 4 choices of +,-,0. Furthermore, whenever any of the nooks below some vertex are assigned 0, all of the nooks below that vertex must be assigned 0. Then a signed dissection of the first kind (which indexes the cells in the simple signed associahedron) requires that only the root vertex can have its nooks assigned 0. And a signed dissection of the second kind (which indexes the cells in the non-simple signed associahedron) requires that only the vertices whose children are all leaves can have their nooks assigned 0. The partial order, roughly speaking, translates into contracting non-leaf edges in the tree and comparing the +,-,0 assignments in the nooks. Do you know of any category/homotopy theoretic applications for either of these signed associahedra? Best wishes, Vic Reiner