A109281 -id:A109281 - 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

A112226 Table T(n,k) of number of elements of Weyl group of type D of order 2^{n-1} n! such that a reduced word uses exactly n-k distinct simple reflections 0 <= k <= n, n>=1.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 13, 7, 3, 1, 135, 40, 12, 4, 1, 1537, 293, 66, 18, 5, 1, 19811, 2646, 451, 100, 25, 6, 1, 289073, 28887, 3753, 663, 143, 33, 7, 1, 4741923, 374820, 37798, 5232, 940, 196, 42, 8, 1, 86705417, 5676121, 457508, 49444, 7174, 1294, 260, 52, 9, 1

Offset: 0

Mike Zabrocki, Aug 28 2005

The first two rows of this table are not well-defined. This is an analog of the notion of permutations with k components for type D (see A059438)

			D_3 is generated by {s_0,s_1,s_2} where s_0^2 = s_1^2 = s_2^2 = (s_0 s_1)^2 = (s_0 s_2)^3 = (s_1 s_2)^2, the elements of this group can be broken up into 4 sets with reduced words as {1}, {s_0, s_1, s_2}, {s_0 s_1, s_1 s_2, s_2 s_1, s_1 s_2 s_1, s_0 s_2, s_2 s_0, s_0 s_2 s_0} hence T(3,3)=1, T(3,2)=3 and T(3,1)=7. T(3,0)=13 since the remaining 13 elements will have reduced words where all three simple reflections appear.

N. Bergeron, C. Hohlweg, M. Zabrocki, Posets related to the connectivity set of Coxeter groups

Cf. A112225, A003319, A109253, A109281, A085771, A059438.

f2:=proc(n,k) local i,gx,g2x; gx:=add(i!*x^i, i=0..n); g2x:=subs(x=2*x,gx); coeff(series(((g2x+3)/(2*gx) + x)*(1-1/gx)^k - x*(1-1/gx)^(k-1),x,n+1),x,n); end: f1:=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); T:=(n,k)->if k=0 then f1(n) else f2(n,k) fi;

max = 10;
fA = 1 - 1/Sum[n!*x^n, {n, 0, max}] + O[x]^max;
fD = (3 + Sum[2^n*n!*x^n, {n, 0, max}])/(2*Sum[n!*x^n, {n, 0, max}]) + x - 2 + O[x]^max;
f = (2*t*fA - 2*t*x + t^2*x*fA + fD)/(1 - t*fA);
row[n_] := CoefficientList[ SeriesCoefficient[f, {x, 0, n}], t];
Join[{{0}}, {{0, 1}}, Table[row[n], {n, 2, max - 1}]] // Flatten (* Jean-François Alcover, Nov 28 2017 *)

G.f.: (g(2x) - (2 t x - 4 t - 2 x + 4) g(x) - 4 t + 3)/(2(t + (1-t) g(x))) where g(x) = sum_{n >= 0} n! x^n o.g.f. for first column given by (g(2x)+3)/(2g(x)) + x - 2 o.g.f. for k^th (k>1) column given by ((g(2x)+3)/(2g(x)) + x)*(1-1/g(x))^{k-1} - x (1-1/g(x))^{k-2}

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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