A260952
Coefficients in asymptotic expansion of the sequences A109253 and A112225.
Original entry on oeis.org
1, -1, -1, -5, -35, -319, -3557, -46617, -699547, -11801263, -220778973, -4532376577, -101246459811, -2444155497191, -63397685488165, -1758278168174137, -51920205021872395, -1626358286062507551, -53865503179448478605, -1880864793407486366353
Offset: 0
A109253(n)/(n!*2^n) ~ (1 - 1/(2*n) - 1/(4*n^2) - 5/(8*n^3) - 35/(16*n^4) - ...
A112225(n)/(n!*2^(n-1)) ~ (1 - 1/(2*n) - 1/(4*n^2) - 5/(8*n^3) - 35/(16*n^4) - ...
A109253
Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections.
Original entry on oeis.org
1, 1, 5, 35, 309, 3287, 41005, 588487, 9571125, 174230863, 3513016445, 77760961991, 1875249535941, 48946667107295, 1374949148971597, 41361812577803383, 1326708910645563669, 45201102932347559503, 1630193308027321807133, 62047171055048539457255
Offset: 0
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.
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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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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 *)
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
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.
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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;
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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 *)
Showing 1-3 of 3 results.
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