A260952 -id:A260952 - cp's OEIS frontend

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

Mike Zabrocki, Aug 19 2005

nonn

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

			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.

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.

Cf. A003319, A109281, A112225, A260952.

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);

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 *)

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

More terms from Vaclav Kotesovec, Aug 05 2015

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.

Original entry on oeis.org

1, 13, 135, 1537, 19811, 289073, 4741923, 86705417, 1752264235, 38832482641, 937035652035, 24465531961465, 687363659349179, 20679220894484897, 663327190230305715, 22600083539456536457, 815088161465498630635

Offset: 2

Mike Zabrocki, Aug 28 2005

nonn

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

			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).

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.

Cf. A003319, A109253, A260952.

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);

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 *)

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

cp's OEIS Frontend

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

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