On some combinatorial identities and harmonic sums

For any m, n epsilon N we first give new proofs for the following well-known combinatorial identities S-n(m) = Sigma(n)(k=1) ((n)(k)) (-1)(k-1) /k(m) = Sigma n >= r1 >= r2 >=. . . >= r(m)>= 1 1 / r1r2 . . . r(m) and Sigma(n)(k=1) (-1)(n-k) ((n)(k)) k(n) = n!, and then we produce the generating function and an integral representation for S-n(m). Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that zeta(3) = 1 / 9 Sigma(infinity)(n=1) H(3)n + 3H(n)H(n)((2)) + 2H(n)((3)) / 2(n), and zeta(5) = 2 / 45 Sigma(infinity)(n=1) H-n(4) + 6H(n)(2)H ((2))(n) + 8H(n)H(n)((3)) + 3(H-n((2)))(2) + 6H(n)((4)) / n2(n), where H-n((i)) are generalized harmonic numbers defined below.