# Signed counting of partition matrices

## TL;DR

This paper establishes a combinatorial equivalence between signed counts of partition matrices and inversion sequences, introduces improper partition matrices, and connects them to Motzkin paths.

## Contribution

It proves a new signed counting identity for partition matrices and introduces improper partition matrices, linking them to Motzkin paths both analytically and bijectively.

## Key findings

- Signed counting of partition matrices equals the size of a subclass of inversion sequences.
- A subset of improper partition matrices is equinumerous with Motzkin paths.
- The equidistribution is established through analytical and bijective methods.

## Abstract

We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' 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.