On a poset of set partitions

We define combinatorially a partial order on the set partitions. 

This order appears to be equivalent to the Bruhat–Chevalley–Renner order on the upper triangular matrices.  

We exhibit a connection between two statistics on set partitions --

the intertwining number (introduced by Richard Ehrenborg and Margaret Readdy in 1998) 

and the depth-index (our parameter which is the rank function of our poset of set partitions). 

This talk is based on joint works with Mahir Bilen Can and Martin Rubey.