Increasing Sequences and Carnegie Mellon

Andrew has 3 trees of height 3 meters, 4 trees of height 4 meters and 6 trees of height 6 meters.

He also owns 7 pavements in 7 different cities which run South-North. He will decorate his pavements with trees. Each pavement must contain one or more of his trees. Andrew is also a Feng-Shui fan, so he believes that no tree on a pavement should be taller than or even equally as tall as a tree to its south.

In how many ways can Andrew decorate his pavements?

For example, a valid decoration is:

Pavement 1: 6

Pavement 2: 3, 6

Pavement 3: 3, 4, 6

Pavement 4: 4, 6

Pavement 5: 4, 6

Pavement 6: 4,6

Pavement 7: 3

The pavements are written in a South-North manner.

Hint: this might be slightly difficult. I suggest first counting the number of decorations where each pavement can contain zero or more trees.