Annular Non-Crossing Matchings

It is well known that the number of distinct non-crossing matchings of n half circles in the half-plane with endpoints on the x-axis equals the nth Catalan number C-n. This paper generalizes that notion of linear non-crossing matchings, as well as the circular non-crossing matchings of Goldbach and Tijdeman, to non-crossings matchings of curves embedded within an annulus. We prove that the number of such matchings vertical bar Ann(n, m)vertical bar with n exterior endpoints and rn interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with C-n = vertical bar Ann(2n + 1, 1)vertical bar. We also develop bijections between specific classes of annular non-crossing matchings and other combinatorial objects such as binary combinatorial necklaces and planar graphs. Finally, we use Burnside's lemma to obtain an explicit formula for vertical bar Ann(n, m)vertical bar for all integers n, m >= 0.