The following example demonstrates how to construct URL API to run a Report Services document with assigned layout in Strategy Web 9.4.x-10.x:
http:///MicroStrategy/asp/Main.aspx? evt=2048001 &src=Main.aspx.2048001 &documentID=9B64FC2540D96A137F4C7BBDE70EF090 &evt=2048093 &src=Main.aspx.rwd.rwframe.rwb.2048093 &SectionKey=K61 &evtorder=2048001%2c2048093 &2048001=1 &2048093=1
evt=2048001 &src=Main.aspx.2048001 &documentID=
&evt=2048093 &src=Main.aspx.rwd.rwframe.rwb.2048093 &SectionKey=
&evtorder=2048001%2c2048093 &2048001=1 &2048093=1

NOTE: Refer to the following technical note to understand how to find layout key: KB38841: How to pass selector answer "All" from targeted hyperlink to a prompt child report in MicroStrategy Web 9.4.x and 10.x
This is a sample of the layout key when viewing in Google Chrome Developer tool:
