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

Enumerative combinatorics on determinants and signed bigrassmannian polynomials