Theory Update 83

Let us obtain the associahedra from the permutohedra, using Postnikov's method. The coefficients $A_{c_1 c_2 \cdots c_n}$ are in fact volumes associated to the simplices $\Delta_{kn}$. These $n!$ objects may be divided into equivalence classes, where $(c_1 c_2 \cdots c_n)$ is equivalent to $(d_1 d_2 \cdots d_n)$ if the length $n+1$ sequences $(\{c_i \},0)$ and $(\{d_i \},0)$ are cyclic shifts of one another. There are always a Catalan number of such equivalence classes, and so they label the vertices of the associahedron. These classes may be selected with the condition

$c_1 + c_2 + \cdots + c_i \geq i$, $\forall i \in 1,2,\cdots,n$

and the volume is then $A_{c_1 c_2 \cdots c_n} = 1^{c_1} 2^{c_2} 3^{c_3} \cdots n^{c_n}$. For example, when $n = 4$ (the familiar dimension $3$ case) we obtain the $14$ volumes

$A_{4000} = 1$, $A_{2110} = 6$, $A_{2101} = 8$, $A_{2011} = 12$,
$A_{1120} = 18$, $A_{1210} = 12$, $A_{1201} = 16$,
$A_{1111} = 24$, $A_{2020} = 9$, $A_{2200} = 4$,
$A_{1300} = 8$, $A_{3100} = 2$, $A_{3010} = 3$, $A_{3001} = 4$

Is it not fascinating how much information the associahedra contain? Astute readers will recognise the basic arithmetic volume formula from the green book.