Tags: #definition

walks on differential posets

Let $P$ be a differential poset. A walk on this poset is a word of ups and downs, starting from the identity $\hat{0}$ and ending at $\hat{0}$.
We may represent this with a word $W$ of $n$ ups and $n$ downs.
For Young's lattice, these are called oscillating tableauxs.

For any word $W$, we denote $f_W$ to be the number of paths in the Hasse diagram of $P$ that start and end at $\hat{0}$ following the pattern of $W$.

Let ${\kappa }_W$ be the Young diagram of the word constructed as follows:

• The diagram is contained within $n\times n$ where $n$ is the number of
• Starting from the bottom left of the $n\times n$ box and parsing $W$ from left to right:
  • for every $U$ in $W$, go right
  • for every $D$ in $W$, go up

Properties

• paths on differential posets and rooks - $f_W=r_W$ where $r_W$ is the number of placements of nonattacking rooks on ${\kappa }_W$

Examples

Let $W=UUDUUDDD$ and $P$ be Fibonacci's lattice.
$f_W=12$.
${\kappa }_W$ is the Young diagram given by $(4,4,4,2)$.
A placement of $n=4$ rooks on ${\kappa }_W$: