A197130 Sum of reflection (or absolute) lengths of all elements in the Coxeter group of type B_n.
1, 10, 100, 1136, 14816, 220032, 3679488, 68548608, 1409347584, 31717048320, 775808778240, 20499651624960, 582040706088960, 17674457139118080, 571655258741145600, 19621314364126003200, 712374154997583052800, 27277192770051951820800
Offset: 1
Examples
a(2)=10 since W(B_2)={1, t_1=s_1, t_2=s_2, t_3=s_1*s_2*s_1, t_4=s_2*s_1*s_2, t_1*t_2=s_1*s_2, t_2*t_1=s_2*s_1, t_1*t_4=s_1*s_2*s_1*s_2} in terms of simple reflections s_1 and s_2.
References
- P. Renteln, The distance spectra of Cayley graphs of Coxeter groups, Discrete Math., 311 (2011), 738-755.
Links
- B. Foster-Greenwood, C. Kriloff, Spectra of Cayley Graphs of Complex Reflection Groups, arXiv preprint arXiv:1502.07392, 2015
Programs
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Maple
seq(2^n*factorial(n)*add((2*k-1)/(2*k),k=1..n),n=1..100);
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Mathematica
Table[2^n*Factorial[n]*Sum[(2*k-1)/(2*k),{k,1,n}],{n,1,100}] -
Sage
[2^n*factorial(n)*sum([(2*k-1)/(2*k) for k in [1..n]]) for n in [1..100]]
Formula
a(n)=Sum_{w in W(B_n)} l_T(w)=|W(B_n)|Sum_{i=1}^n (d_i-1)/d_i=2^n*n!*(1/2+3/4+...+(2n-1)/(2n)) where T=all reflections in W(B_n), l_T(1)=0 and otherwise l_T(w)=min{k|w=t_1*...*t_k for t_i in T}, and d_1,...,d_n are the degrees of W(B_n)