# Enumerative combinatorics on determinants and signed bigrassmannian polynomials

## TL;DR

This paper introduces signed bigrassmannian polynomials and a bigrassmannian determinant, providing new combinatorial tools and determinantal formulas related to permutations and Bruhat order.

## Contribution

It presents the first definitions of signed bigrassmannian polynomials and the bigrassmannian determinant, expanding enumerative combinatorics with new algebraic and determinant-based methods.

## Key findings

- Bigrassmannian determinant satisfies weighted condensation.
- Determinantal expression for signed bigrassmannian polynomials.
- Introduces a $q$-analog of the determinant related to permutation statistics.

## Abstract

As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a $q$-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).