This week i tried and practice the way to calculate double schubert polynomials with a square tiling. even though i couldnt connect the tilings i counted some of the blank squares in even and odd matrices with 2 particular tiles in the bottom right corner, the cross tile and one of the elbow tiles. I think they are the only ones to do it with. Anyways I had another condition to add only cross tiles in the diagonal aside from the 2 corner pieces i mentioned. so i tried and count how many rows and columns were necessary to complete the tiling if the farthest out cross tile from the corner exits through the closest edges and the closer tilings also exit through the next closest ones- the closer to the corner the further out they exit through the top part of the crosses. in the graphic i made its easier to see. the cross has two parts that exit: two edges that leave through the bottom and side and the side of the cross pointed toward the opposite corner exits a few rows and columns in the other direction. it took a lot of attempts. after i tried and alter the cross tiles in the diagonal to an elbow tile and it doesnt leave all the edges with a line exiting them so i forgot about it for now.
in the odd nxn squares and even n-1 x n-1 squares there is the same number of blank squares. the odd squares blank can be moved to the last indices possible of the even squares- aside from that i can not really tell how a square nxn tiling and square n-1 x n-1 tiling are supposed to calculate two different schubert polynomials.
here is another graphic with a 3 by 3 that has the outer blank diagonal shifted in to indices 2,2 beside a 2 by 2 without the possibility of shifting the blank square in at all. That is my best guess why the different sizes are different polynomials. can anybody explain more precisely?
Technically there's an object that finds the coefficient of a term in power series', there should be one that finds the available indices of blank squares.
In another question about Stanley introduction to hyperplane arrangements multiple types of coefficients of the variable term $t^{k}$ are able to be expressed by $[t^{k}] \chi (t)$.