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
Keywords
Examples
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.
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.
Programs
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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);
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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 *)
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
Extensions
More terms from Vaclav Kotesovec, Aug 05 2015
Comments