As difficult as 01 10 11

\[ \begin{eqnarray} && 9 \\ && 1 + 8 \\ && 2 + 7 \\ && 3 + 6 \\ && 4 + 5 \\ && 1 + 2 + 6 \\ && 1 + 3 + 5 \\ && 2 + 3 + 4 \\ \end{eqnarray} \]

Above shows a total of eight number of ways to express the number 9 as the sum of distinct positive integer in ascending order.

What is the total number of ways to express the number 58 as the sum of distinct positive integer in ascending order?

Bonus: Define \(f(n) \) as the total number of ways to express the positive integer \(n\) as the sum of distinct positive integers in ascending order. Prove that there exist a polynomial of minimum degree \( \frac {n(n+1)}2 \) such that you can find \(f(1), f(2), \ldots, f(n) \). Construct this polynomial.

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