engleski

Realizations of Stasheff Polytope by Means of Alternating Sign Matrices

A method of determinant evaluation known as the condensation has a natural interpretation in terms of the alternating sign matrices. An alternating sign matrix (ASM) is a square matrix of 0s, 1s, and -1s such that the sum of each row and each column is 1 and the nonzero entries in each row and each column alternate in sign. These matrices generalize permutation matrices. We study a family of ASMs with pattern avoidance and the property that the rightmost 1 in the row i+1>=2 is to the right of the leftmost 1 in the row i, and further families with analogous properties. We will show symmetries among these families of matrices. Stasheff’s polytope, also known as asociahedron, appeared in Jim Staheff’s work in the 1960s. A d-asociahedron is a convex polytope of dimension d whose i-dimensional sides are denoted by significant orders of d- i pairs of parentheses between d + 2 independent variables with appropriate incidence mapping. The vertices of the associahedron can also be joined by triangulation of a regular polygon. We establish a one to one correspondence between the family of AMS of order d + 2 and vertices of d-asociahedron.

Stasheff polytope, Alternating Sign Matrices

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