Short generating functions for some semigroup algebras
The electronic journal of combinatorics, Tome 10 (2003)
Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.
@article{10_37236_1729,
author = {Graham Denham},
title = {Short generating functions for some semigroup algebras},
journal = {The electronic journal of combinatorics},
year = {2003},
volume = {10},
doi = {10.37236/1729},
zbl = {1023.05008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1729/}
}
Graham Denham. Short generating functions for some semigroup algebras. The electronic journal of combinatorics, Tome 10 (2003). doi: 10.37236/1729
Cité par Sources :