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%I #22 Mar 17 2020 11:57:34
%S 1,3,35,307,218431,69851351,37931027461
%N Numerator of the harmonic numbers for the symmetric group relative to the Coxeter length.
%C This is a generalization of the harmonic numbers to finite groups:
%C Let G be a finite group, S <= G a generating set. Set H(G) := Sum_{g in G} 1/(|g|+1), where |g|:= word length (with respect to S). For G = SymmetricGroup(n) and |.| = Coxeter length we get H(G) = a(n) / b(n), where the sequence b(n) = denominator of H(G) will be defined elsewhere.
%C Relation to Lagarias inequality (which is equivalent to Riemann Hypothesis), G = C_n = cyclic group, S = {+1}: sigma(G) <= H(G) + exp(H(G))*log(H(G)) (conjecture, which implies RH).
%C Relation to harmonic numbers: H_n = H(C_n), where C_n = cyclic group.
%H MathOverflow, Definition of the harmonic numbers for each finite group, <a href="https://mathoverflow.net/questions/330077/a-group-theoretic-interpretation-of-lagarias-inequality"> A Group theoretic interpretation of Lagarias inequality</a>
%F H(G) := Sum_{g in G} 1/(|g|+1), G a finite group, S<= G a generating set, |g| := word length relative to S.
%e For n=1..7 the harmonic numbers relative to the Coxeter length in the Symmetric group S_n are given by: 1, 3/2, 35/12, 307/42, 218431/9240, 69851351/720720, 37931027461/77597520, hence a(1) = 1, a(2) = 3, a(3) = 35 etc.
%o (Sage)
%o def HG(G):
%o return sum(1 / (g.length() + 1) for g in G)
%o [HG(SymmetricGroup(n)).numerator() for n in range(1, 8)]
%K nonn,frac,more
%O 1,2
%A _Orges Leka_, May 10 2019
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