6127: Greg and Caves

Greg has a pad. The pad's screen is an n×m rectangle, each cell can be either black or white. We'll consider the pad rows to be numbered with integers from 1 to n from top to bottom. Similarly, the pad's columns are numbered with integers from 1 to m from left to right.

Greg thinks that the pad's screen displays a cave if the following conditions hold:

• There is a segment [l,r] (1≤l≤r≤n), such that each of the rows l,l+1,...,r has exactly two black cells and all other rows have only white cells.
• There is a row number t (l≤t≤r), such that for all pairs of rows with numbers i and j (l≤i≤j≤t) the set of columns between the black cells in row i (with the columns where is these black cells) is the subset of the set of columns between the black cells in row j (with the columns where is these black cells). Similarly, for all pairs of rows with numbers i and j (t≤i≤j≤r) the set of columns between the black cells in row j (with the columns where is these black cells) is the subset of the set of columns between the black cells in row i (with the columns where is these black cells).

Greg wondered, how many ways there are to paint a cave on his pad. Two ways can be considered distinct if there is a cell that has distinct colors on the two pictures.

Help Greg.