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I.K
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Combinatorial identity involving number of cycles (of any length) in a permutation
I see that now. From the context, it's clear he means the usual: $$ \prod_{i=0}^{n-1}(\beta+i) = \sum_{\sigma \in S_n}\beta^{c(\sigma)} $$ as Linus noted. Thanks everyone for the quick help!
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Combinatorial identity involving number of cycles (of any length) in a permutation
Thank you very much! This also clears up some later applications in the paper. This is just the result in Stanley V1 p. 27 (Prop. 1.3.7)
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Combinatorial identity involving number of cycles (of any length) in a permutation
Thank you. I'm going through it by hand for n=1,2,3.. and it does seem that it should be + and not -
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Combinatorial identity involving number of cycles (of any length) in a permutation
Yes. I rewrote it. I used $ \beta = \alpha^{-1} $ Then it's the same equality, isn't it?
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Combinatorial identity involving number of cycles (of any length) in a permutation
I edited the question, I'm pretty sure that from the context, beta is positive strictly less than 1.
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Combinatorial identity involving number of cycles (of any length) in a permutation
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Combinatorial identity involving number of cycles (of any length) in a permutation
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Combinatorial identity involving number of cycles (of any length) in a permutation