Speaker : Prof. Zhicheng Gao (University of California)
Title : Asymptotic properties of compositions over finite groups.
Time : 2019-01-15 (Tue) 14:00 -
Place : Seminar Room 638, Institute of Mathematics (NTU Campus)
Abstract: Let $\Gamma$ be a finite additive group. An $m$-composition over $\Gamma$ is an $m$-tuple $(g_1,g_2,\ldots,g_m)$ over $\Gamma$. It is called an $m$-composition of $g$ if $\sum_{j=1}^m g_j = g$. A composition $(g_j)$ over $S$ is called {\em locally restricted} if there is a positive integer $\sigma$ such that any $\sigma$ consecutive parts of $(g_j)$ satisfy certain conditions. Locally restricted compositions over $\Gamma$ are associated with walks in a de Bruijin graph. Under certain aperiodic conditions, we will show that the asymptotic number of $m$-compositions of $\gamma$ is independent of $\gamma$. We also show that the distribution of the number of occurrences of a set of subwords in such $m$-compositions is asymptotically normal with mean and variance proportional to $m$. The proofs use the transfer matrix, Kronecker product of matrices, and Perron-Frobenius theorem.