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A096364
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Number of ways to generate a Coxeter element of the reflection group of the root system B_n with certain restrictions on generators: (3n-4)*(n-2)^(n-2) - (n-1)^(n-1).
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0
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0, 1, 1, 5, 41, 459, 6469, 109577, 2164273, 48787127, 1235194181, 34688329389, 1069808023129, 35936710441475, 1305872544724357, 51034409943693329, 2134268774190839009, 95096941799140816623, 4497325804679310925957
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OFFSET
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1,4
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COMMENTS
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Let G be the group (Z/2Z)^n * S_n. (Here * denotes the semidirect product. Each member of G is a function f from {-n, -n+1, ..., -2, -1, 1, 2, ..., n-1, n} to itself such that f is bijective and f(-x) = -f(x). The operation is composition.) Let C be a Coxeter element of G such that C(i) = i+1 for i = 1, 2, ..., n-1 and C(n) = -1. Let s_ij, i eq j and i,j > 0, be the function such that s_ij(i) = j, s_ij(j) = i and s_ij(k) = k for all k eq i,j, k > 0. Let s_2 be the function such that s_2(2) = -2 and s_2(k) = k for all k eq 2. Then a(n) is the number of ways to write C as a product of the s_ij's and s_2, using as few elements as possible.
a(n) is the number of ways to write the Coxeter element s_{e_1}s_{e_1-e_2}s_{e_2-e_3}s_{e_3-e_4}...s_{e_{n-1}-e_n} of the reflection group of the root system B_n as a product of reflections from {s_{e_i - e_j}, e_2}, using as few reflections as possible.
a(n) is also the number of ways to write the same Coxeter element as a product of reflections from {s_{e_i - e_j}, e_2} - {s_{e1-e2}} \union {s_{e1+e2}}, using as few reflections as possible.
Let T be a spanning tree in complete graph K_n on n labeled nodes. Let w(T) be the least neighbor of n in T. Let A_n be the set of spanning tree T of K_n such that the least neighbor or n is a leaf. Then for n >= 3, sum_{T \in A_n} w(T) = a(n).
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REFERENCES
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Richard Kane, Reflection Groups and Invariant Theory, Springer-Verlag, New York, 2001; 9-10.
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LINKS
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FORMULA
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a(n) = (3n-4)*(n-2)^(n-2) - (n-1)^(n-1) for n >= 3.
E.g.f.: (x*(x-1)*LambertW(-x)-x^2)/LambertW(-x)^2. - Vladeta Jovovic, Jul 02 2004
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EXAMPLE
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a(4)=5 because we can write s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4} =
1) s_{e1-e2}s_{e2}s_{e2-e3}s_{e3-e4}
2) s_{e1-e2}s_{e2}s_{e2-e4}s_{e2-e3}
3) s_{e1-e2}s_{e2}s_{e3-e4}s_{e2-e4}
4) s_{e1-e2}s_{e3-e4}s_{e2}s_{e2-e4}
5) s_{e3-e4}s_{e1-e2}s_{e2}s_{e2-e4}
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MATHEMATICA
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a[1] = 0; a[2] = a[3]1; a[n_] := (3n - 4)*(n - 2)^(n - 2) - (n - 1)^(n - 1); Table[ a[n], {n, 19}] (* Robert G. Wilson v, Jul 24 2004 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Pramook Khungurn (pramook(AT)mit.edu), Jul 01 2004
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EXTENSIONS
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STATUS
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approved
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