A308033 Numerator of the harmonic numbers for the symmetric group relative to the Coxeter length.

1, 3, 35, 307, 218431, 69851351, 37931027461

Offset

1,2

Comments

This is a generalization of the harmonic numbers to finite groups:

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.

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).

Relation to harmonic numbers: H_n = H(C_n), where C_n = cyclic group.

Links

Table of n, a(n) for n=1..7.

MathOverflow, Definition of the harmonic numbers for each finite group, A Group theoretic interpretation of Lagarias inequality

Formula

H(G) := Sum_{g in G} 1/(|g|+1), G a finite group, S<= G a generating set, |g| := word length relative to S.

Example

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.

Prog

(Sage)
def HG(G):
    return sum(1 / (g.length() + 1) for g in G)
[HG(SymmetricGroup(n)).numerator() for n in range(1, 8)]

Crossrefs

Sequence in context: A339516 A121078 A289950 * A198961 A112488 A221922

Adjacent sequences: A308030 A308031 A308032 * A308034 A308035 A308036

Keyword

nonn,frac,more

Author

Orges Leka, May 10 2019

Status

approved