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Math Help - Proving an identity with unsigned sterling numbers of first kind

  1. #1
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    Proving an identity with unsigned sterling numbers of first kind

    How do I prove c(n+1, m+1) = \sum_{k=0}^{n} c(n, k) \binom{k}{m}
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    Re: Proving an identity with unsigned sterling numbers of first kind

    have you tried induction?
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    Re: Proving an identity with unsigned sterling numbers of first kind

    The base case is easy to show, but for the induction step do I assume c(n, m+1) = \sum_{k=0}^{n-1} c(n-1, k) \binom{k}{m} is true? and then try to get to c(n+1, m+1) = \sum_{k=0}^{n} c(n, k) \binom{k}{m}?

    Edit: I am assuming that we want to induct on n.
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    Re: Proving an identity with unsigned sterling numbers of first kind

    Quote Originally Posted by evg952 View Post
    The base case is easy to show, but for the induction step do I assume c(n, m+1) = \sum_{k=0}^{n-1} c(n-1, k) \binom{k}{m} is true? and then try to get to c(n+1, m+1) = \sum_{k=0}^{n} c(n, k) \binom{k}{m}?

    Edit: I am assuming that we want to induct on n.
    err what's the difference between $c(m,n)$ and $\begin{pmatrix}m \\n\end{pmatrix}$ ?
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