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

2,2

Comments

This is an analog for type D of the concept of connected permutations (see A003319 and A109253).

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.

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

Example

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).

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);

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 *)

Crossrefs

Cf. A003319, A109253, A260952.

Sequence in context: A179619 A019519 A243216 * A069511 A052262 A065550

Adjacent sequences: A112222 A112223 A112224 * A112226 A112227 A112228

Keyword

nonn

Author

Mike Zabrocki, Aug 28 2005

Status

approved