Merge branch 'public/combinat/bounded_affine_permutation-22187' in 8.…

…0.b8

src/doc/en/reference/references/index.rst

@@ -993,6 +993,11 @@ REFERENCES:
             Sci. Publ., Hackensack, NJ, 2008. Preprint version:
             :arxiv:`0710.1835`
 
+.. [KLS2013] Allen Knutson, Thomas Lam, and David Speyer.
+             *Positroid Varieties: Juggling and Geometry*
+             Compositio Mathematica, **149** (2013), no. 10.
+             :arXiv:`1111.3660`.
+
 .. [KMM2004] Tomasz Kaczynski, Konstantin Mischaikow, and Marian
              Mrozek, "Computational Homology", Springer-Verlag (2004).
 

src/sage/combinat/permutation.py

@@ -182,6 +182,7 @@
     :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:`bounded_affine_permutation` | Return 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.
@@ -6986,6 +6987,57 @@ def bistochastic_as_sum_of_permutations(M, check = True):
 
     return value
 
+def bounded_affine_permutation(A):
+    r"""
+    Return the bounded affine permutation of a matrix.
+
+    The *bounded affine permutation* of a matrix `A` with entries in `R`
+    is 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, \ldots, j`,
+    over `R`, where column indices are taken modulo `n`.
+    If column `i` is the zero vector, then the permutation has a
+    fixed point at `i`.
+
+    INPUT:
+
+    - ``A`` -- matrix with complex entries
+
+    EXAMPLES::
+
+        sage: from sage.combinat.permutation import bounded_affine_permutation
+        sage: A = Matrix(QQ, [[1,0,0,0], [0,1,0,0]])
+        sage: bounded_affine_permutation(A)
+        [5, 6, 3, 4]
+
+    REFERENCES:
+
+    - [KLS2013]_
+    """
+    n = A.ncols()
+    R = A.base_ring()
+    from sage.modules.free_module import FreeModule
+    from sage.modules.free_module import span
+    z = FreeModule(R, A.nrows()).zero()
+    v = A.columns()
+    perm = []
+    for j in range(n):
+        if v[j] == z:
+            perm.append(j+1)
+            continue
+        V = span([z], R)
+        for i in range(j+1, j+n+1):
+            index = i % n
+            V = V + span([v[index]], R)
+            if V == span([z], R):
+                continue
+            if v[j] in V:
+                perm.append(i+1)
+                break
+    S = Permutations(2*n, n)
+    return S(perm)
+
+
 class StandardPermutations_descents(StandardPermutations_n_abstract):
     """
     Permutations of `\{1, \ldots, n\}` with a fixed set of descents.