Update on my evil math problem
Slightly redefining the problem to make it slightly less evil, I now have:
For arbitrary m and n, let there be n-dimensional vectors X for whole number coordinates x1, x2, ... xn. What is the size of the set S which includes all X such that the sum from i=1 to i=n of xi is equal to m?
I have solutions which are general for n greater than or equal to m for m less than or equal to 6:
(abbreviating "n, choose" as nc)
m=0: nc0
m=1: nc1
m=2: nc1 + nc2
m=3: nc1 + 2*nc2 + nc3
m=4: nc1 + 3*nc2 + 3*nc3) + nc4
m=5: nc1 + 4nc2 + 6nc3+ 4nc4 + nc5
m=6: nc1 + 5nc2 + 10nc3 + 10nc4 + 5nc5 + nc6
I'm definitely struggling to see a general form for the pattern. Anybody enough of a math geek to have ideas?
For arbitrary m and n, let there be n-dimensional vectors X for whole number coordinates x1, x2, ... xn. What is the size of the set S which includes all X such that the sum from i=1 to i=n of xi is equal to m?
I have solutions which are general for n greater than or equal to m for m less than or equal to 6:
(abbreviating "n, choose" as nc)
m=0: nc0
m=1: nc1
m=2: nc1 + nc2
m=3: nc1 + 2*nc2 + nc3
m=4: nc1 + 3*nc2 + 3*nc3) + nc4
m=5: nc1 + 4nc2 + 6nc3+ 4nc4 + nc5
m=6: nc1 + 5nc2 + 10nc3 + 10nc4 + 5nc5 + nc6
I'm definitely struggling to see a general form for the pattern. Anybody enough of a math geek to have ideas?