Quivers (directed graphs) were introduced to specify a class of quantum field theories (QFTs), relevant to "D-branes" in string theory. Permutation methods have been used to solve counting problems of interest in these QFTs. The solution reveals surprising underlying structures related to word combinatorics. These connections arise when the number N of D-branes is large. For more general N, the counting problems can be expressed in terms of Littlewood-Richardson coefficients. Further afield, these coefficients have been the subject of recent research in connection with computational complexity.