Front representation of set partitions

Let $π$ be a set partition of $[n]=\{1,2,...,n\}$. The standard representation of $π$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer between $i$ and $j$. The front representation of $π$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is $i$. Using the front representation, we find a recurrence relation for the number of $12... k12$-avoiding partitions for $k\geq2$. Similarly, we find a recurrence relation for the number of $k$-distant noncrossing partitions for $k=2,3$. We also prove that the front representation has several joint symmetric distributions for crossings and nestings as the standard representation does.

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