The Number of Admissible Sequences for Indecomposable Serial Rings

Abstract

We give a formula for the number of admissible sequences for indecomposable serial rings with $n$ indecomposable projective modules whose minimum composition length is less than or equal to $m$. In particular, if $n=m$ is prime, we show that the number of such admissible sequences is $${{2n-1}\choose n} +{1\over n} \bigg[ (n-1)^2 - {{2n-2}\choose {n}}\bigg] .$$