Thursday, April 21, 2011

Theory Update 80

Postnikov writes the formula for the number of parking functions $P_n$ in terms of trees on $n$ nodes. Thus for $n = 3$ we draw five trees, and obtain $16$ possible elements from the formula

$(n + 1)^{n - 1} = \sum_{T} \frac{n!}{2^{n}} \prod_{v} (1 + \frac{1}{h(v)})$

where the sum is over trees $T$, the product over nodes $v$ in a tree, and $h(v)$ is the hook length for a given node. Observe that the five trees resemble the internal part of a four leaf tree. These are the trees that label the pentagon associahedron. Thus the $16$ simplices of an associahedron in three dimensions are labeled by the lower dimensional associahedron.

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