A109253 Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections.

1, 1, 5, 35, 309, 3287, 41005, 588487, 9571125, 174230863, 3513016445, 77760961991, 1875249535941, 48946667107295, 1374949148971597, 41361812577803383, 1326708910645563669, 45201102932347559503, 1630193308027321807133, 62047171055048539457255

Offset

0,3

Comments

This is the analog of a connected permutation (permutation with no global ascent) in type B.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..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

O.g.f.: g(2x)/g(x) where g(x) = sum_{n>=0} n! x^n.

a(n) ~ n! * 2^n * (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 28 2015

Example

For n=2, the Weyl group B_2 has 8 elements and is generated by {t,s} with s^2=t^2=(st)^4=1, the elements which have reduced words containing both s and t are st, ts, sts, tst and stst. The other three elements are 1, s, t. Therefore f(2)=5.

Maple

f:=k->coeff(series(add(2^n*n!*x^n, n=0..k)/add(n!*x^n, n=0..k), x, k+1), x, k);

Mathematica

nmax = 20; CoefficientList[Assuming[Element[x, Reals], Series[1/2*Exp[1/(2*x)] * ExpIntegralEi[1/(2*x)] / ExpIntegralEi[1/x], {x, 0, nmax}]], x] (* Vaclav Kotesovec, Aug 05 2015 *)

Crossrefs

Cf. A003319, A109281, A112225, A260952.

Sequence in context: A253096 A305964 A226739 * A052797 A371540 A225177

Adjacent sequences: A109250 A109251 A109252 * A109254 A109255 A109256

Keyword

nonn

Author

Mike Zabrocki, Aug 19 2005

Extensions

More terms from Vaclav Kotesovec, Aug 05 2015

Status

approved