inital commit for adding bdd_affine_perm method in permutation

src/sage/combinat/permutation.py

@@ -182,13 +182,13 @@
     :meth:`from_lehmer_code` | Returns the permutation with Lehmer code ``lehmer``.
     :meth:`from_reduced_word` | Returns the permutation corresponding to the reduced word ``rw``.
     :meth:`bistochastic_as_sum_of_permutations` | Returns a given bistochastic matrix as a nonnegative linear combination of permutations.
+    :meth:`bdd_affine_perm` | Returns a partial permutation representing the bounded affine permutation of a matrix
     :meth:`descents_composition_list` | Returns a list of all the permutations in a given descent class (i. e., having a given descents composition).
     :meth:`descents_composition_first` | Returns the smallest element of a descent class.
     :meth:`descents_composition_last` | Returns the largest element of a descent class.
     :meth:`bruhat_lequal` | Returns ``True`` if ``p1`` is less or equal to ``p2`` in the Bruhat order.
     :meth:`permutohedron_lequal` | Returns ``True`` if ``p1`` is less or equal to ``p2`` in the permutohedron order.
-    :meth:`to_standard` | Returns a standard permutation corresponding to the permutation ``self``.
-
+    :meth:`to_standard` | Returns a standard permutation corresponding to the permutation ``self``
 AUTHORS:
 
 - Mike Hansen
@@ -242,7 +242,7 @@
 from sage.structure.list_clone import ClonableArray
 from sage.structure.global_options import GlobalOptions
 from sage.interfaces.all import gap
-from sage.rings.all import ZZ, Integer, PolynomialRing
+from sage.rings.all import ZZ, Integer, PolynomialRing, CC
 from sage.arith.all import factorial
 from sage.matrix.all import matrix
 from sage.combinat.tools import transitive_ideal
@@ -260,6 +260,8 @@
 from sage.combinat.rsk import RSK, RSK_inverse
 from sage.combinat.permutation_cython import (left_action_product,
              right_action_product, left_action_same_n, right_action_same_n)
+from sage.modules.free_module_element import zero_vector
+from sage.modules.free_module import span
 
 class Permutation(CombinatorialElement):
     r"""
@@ -6985,6 +6987,60 @@ def bistochastic_as_sum_of_permutations(M, check = True):
         value += minimum * CFM(P([x[1]-n+1 for x in matching]))
 
     return value
+
+
+def bdd_affine_perm(A):
+    r"""
+    Return the bounded affine permutation of a matrix.
+
+
+    INPUT:
+
+    -"A"--a matrix with complex entries.
+
+    OUTPUT:
+
+    A partial permutation of length n, where n is the number of columns of A.
+    The entry in position i is the smallest value j such that
+    column i is in the span of columns i+1,...,j, over the complex numbers,
+    where column indices are taken modulo n.  
+    If column i is the zero vector, then the permutation has a fixed point at i. 
+
+    EXAMPLES::
+        sage: from sage.combinat.permutation import bdd_affine_perm
+        sage: A = Matrix(CC,[[1,0,0,0],[0,1,0,0]])
+        sage: bdd_affine_perm(A)
+        [5, 6, 3, 4]
+
+    REFERENCES:
+
+    For more on bounded affine permutations, see
+
+    ..[KLS] Allen Knutson, Thomas Lam, and David Speyer.
+    Positroid Varieties: Juggling and Geometry
+    Compositio Mathematica, Volume 149, Issue 10, 2013
+    arXiv:1111.3660 [math.AG]
+    """
+    n = A.ncols()
+    z = zero_vector(CC, A.nrows())
+    v = A.columns()
+    perm = []
+    for j in range(0,n):
+        if v[j] == z:
+            perm.append(j+1)
+            continue
+        V = span([z],CC)
+        for i in range(j+1,j+n+1):
+            index = i % n
+            V = V + span([v[index]],CC)
+            if V == span([z],CC):
+                continue
+            if v[j] in V:
+                perm.append(i+1)
+                break
+    S = Permutations(range(1, 2 * n+1),n)
+    return S(perm)
+
 
 class StandardPermutations_descents(StandardPermutations_n_abstract):
     """