Inversion formula for the growth function of a cancellative monoid

Details Inversion formula for the growth function of a cancellative monoid Inversion formula for the growth function of a cancellative monoid

2012-01-26

arxiv.org/abs/1201.5496v4

Mathematics

Combinatorics

18 pages

Scientific paper

We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge0}$ and associated generating function $P(t)=\sum_{m\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skew-growth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(-s)$, the inversion formula turns out to be the Euler product formula for the Riemann's zeta function.

Affiliated with

Also associated with

No associations

Rating

Inversion formula for the growth function of a cancellative monoid does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Inversion formula for the growth function of a cancellative monoid, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Inversion formula for the growth function of a cancellative monoid will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-445450