Department of Mathematics, Faculty of Science, Okayama University
Doi
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)
