Decorated permutations

Returns a matrix, each row of which is a permutation of the numbers 1:N wherein fixed elements take one of two signs.

D = decorated_perms(N) returns a matrix, each row of which is a permutation of the numbers 1:N, within which fixed elements take one of two
orientations for each fixed point (loop): clockwise (positive) or counterclockwise (negative).

D=decorated_perms(N,K) returns only those decorated permutations for which the sum of the number of exceedances and the number of counterclockwise loops is equal to K. In this case, the number of rows of D is the number of totally positive cells in the positive Grassmannian
Gr^+(n,k). For more information see Williams, L.K., 2005. The figure is reproduced from Ardila et al. 2014.

Williams, L.K., 2005. Enumeration of totally positive Grassmann cells. Advances in Mathematics, 190(2), pp.319-342.

Ardila, F., Rincón, F. and Williams, L., 2014. Positroids, non-crossing partitions, and positively oriented matroids. In Discrete Mathematics and Theoretical Computer Science (pp. 655-666). Discrete Mathematics and Theoretical Computer Science.

The figure is frmo