Let h(n,d) be # of ways of writing the identity in S_d as a product of n transpositions. Define generating function G(x,y)=∑h(n,d)/(n!d!) (n, d=0...). Let H(x,y)=log(G(x,y)). Then H(x,y) satisfies: D^2 H(x,y) = y exp(H(x,y exp(x))-2H(x,y)+H(x,y exp(-x))) where D=y∂/∂y
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Replying to @sigfpe
G(x,y)=∑h(n,d)/(n!d!) (n, d=0...). should probably be G(x,y)=∑h(n,d) x^n y^n / (n!d!) (n, d=0...) Right? I'd like to understand this better. The logarithm usually comes from looking at "connected" things, like connected Feynman diagrams.
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Replying to @johncarlosbaez @sigfpe
Actually probably more like G(x,y)=∑h(n,d) x^n y^d / (n!d!) (n, d=0...), I imagine...
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Replying to @johncarlosbaez @sigfpe
So easy to make these tiny slips.
9:21 AM - 28 Jan 2020
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