#OK to post homework
#Joseph Koutsoutis, 04-27-2025, Assignment 25

read `C25.txt`:

#3

NuEqC := proc(n, c) local x, p, i:
  p := CIPsn(n, x):
  subs({seq(x[i] = c, i=1..n)}, p):
end:

#[seq(NuEqC(i,2),i=1..12)] outputs [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

#[seq(NuEqC(i,3),i=1..12)] outputs [3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91]

#[seq(NuEqC(i,4),i=1..12)] outputs [4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455]

#We observe that for these values, the output matches [seq(binomial(i+c-1, c-1), i=1..12)].
#Since we are looking at the action of the full symmetric group on sequences of length n with
#entries in [c], we can interpret this as finding how many ways we can color n indistinguishable 
#objects using c colors. By stars and bars, there are binomial(n+c-1, c-1) ways to do this.