# Combinatorics (splitting integral)

I am trying to split an integral in a way that the integrating region becomes "symmetric" i.e. for a fixed $p=1,\dots,m$, $$\int_{s<u_1<\cdots u_m<t}\int_{r<v_1<\cdots< v_n<u_p} dv_1\cdots dv_n du_1\cdots du_m = \sum \int_{r\wedge s<w_1<\dots w_{n+m}<t} dw_1\cdots dw_{n+m}$$ I claim this is possible! But some combinatorics come up! I wonder how many terms there are in the sum.

It seems to me like computing "how many ways there are to order $n$ variables among the first $s,u_1,\dots,u_p$. Is this right?

Thank you very much for your help! :)

• I think I solved it: One can easily split it in $p$ terms: $\sum_{j=0}^{p-1} \int_{r\wedge s < u_1 < \cdots u_j<v_1<\cdots v_n<u_{j+1}<\cdots <u_m<t}$ taking $u_0:=r\wedge s$ So there are $p$ sums. – Dan Feb 9 '13 at 21:20
• Please answer your own question, and accept it to close the issue. – vonbrand Feb 9 '13 at 21:46
• I can't do it before 5 hours from now.. – Dan Feb 9 '13 at 22:02