A112225 Number of elements of a Weyl group of order 2^{n-1} n! of type D for which a reduced word contains all of the simple reflections.
1, 13, 135, 1537, 19811, 289073, 4741923, 86705417, 1752264235, 38832482641, 937035652035, 24465531961465, 687363659349179, 20679220894484897, 663327190230305715, 22600083539456536457, 815088161465498630635
Offset: 2
Keywords
Examples
For n=2, the Weyl group of order 4 is generated by {s_0', s_1} with (s_0')^2=s_1^2 = (s_0' s_1)^2 = 1, s_0' s_1 is the only element with a reduced word containing both simple reflections (the other elements are 1, s_0' and s_1).
For n=3, the Weyl group of type D is isomorphic to S_4 where there are 13 'connected permutations' (see A003319).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 2..400
- N. Bergeron, C. Hohlweg, M. Zabrocki, Posets related to the connectivity set of Coxeter groups, arXiv:math/0509271 [math.CO], 2005-2006.
- Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Programs
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Maple
f:=n->coeff(series((add(2^k*k!*x^k,k=1..n)+4)/add(2*k!*x^k,k=0..n)+x-2,x,n+1),x,n);
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Mathematica
nmax = 20; Rest[Rest[CoefficientList[Assuming[Element[x, Reals], Series[(Exp[1/(2*x)] * ExpIntegralEi[1/(2*x)] + 6*x*Exp[1/x]) / (4*ExpIntegralEi[1/x]) + x - 2, {x, 0, nmax}]], x]]] (* Vaclav Kotesovec, Aug 05 2015 after Martin and Kearney *)
Formula
G.f.: f(x) = (g(2x)+3)/(2 g(x)) + x - 2 where g(x) = sum_{n >= 0} n! x^n.
a(n) ~ n! * 2^(n-1) * (1 - 1/(2*n) - 1/(4*n^2) - 5/(8*n^3) - 35/(16*n^4) - 319/(32*n^5) - 3557/(64*n^6) - 46617/(128*n^7) - 699547/(256*n^8) - 11801263/(512*n^9) - 220778973/(1024*n^10)), for coefficients see A260952. - Vaclav Kotesovec, Jul 29 2015
Comments