in reply to Re^2: Combinatorics problem. (Updated with more info.)
in thread Combinatorics problem. (Updated with more info.)

From DLMF or MathWorld we see the number of restricted compositions is binomial(n-1,m-1). So:

perl -Mntheory=:all -E 'say binomial(12-1,7-1)' 462

Basically the same with Math::Pari or Pari/GP.

Chinn and Heubach (2003) and Opdyke (2008) have interesting info relating restricted compositions to Pascal's triangle.

Edit: There's also some interesting but scattered info on OEISWiki, including the idea of encoding the compositions as RLE on binary. For generating the sequences, Kelleher's paper is a great reference.