A major index for matchings and set partitions

William Y. C Chen, Ira M. Gessel, Catherine H. Yan, Arthur L. B. Yang

Published 2007-01-03Version 1

We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major index.