# Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function. There exists a nice product formula for the principal specializations:

sλ(1, q, q2, ..., qn-1) = Πi<j (qλi+n-i - qλj+n-j) / (qj-1 - qi-1).

Is a similar evaluation known for specializations of the type sλ(1, 2, ..., n)?

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Two quick observations: for lambda = (1, 1, ...), e_k(1, 2, ... n) is an unsigned Stirling number of the first kind, and for lambda = (k), h_k(1, 2, ... n) is a Stirling number of the second kind. –  Qiaochu Yuan Oct 15 '09 at 18:38
It does seem to be the case that it's always a rational function in $n$. Is this obvious? –  Martin Rubey Mar 16 '10 at 20:49