Signed counting of partition matrices

Abstract

We prove that the signed counting (with respect to the parity of the ``$\mathrm{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively.

28 pages. Most of Section 4 has been rewritten, with more questions raised in Outlook