def to_pair_of_twin_binary_trees(self, p):
r"""
Apply a bijection between Baxter permutations of size ``self._n``
and the set of pairs of twin binary trees with ``self._n`` nodes.
INPUT:
- ``p`` -- a Baxter permutation.
OUTPUT:
The pair of twin binary trees `(T_L, T_R)` where `T_L`
(resp. `T_R`) is obtained by inserting the letters of ``p`` from
left to right (resp. right to left) following the the binary search
tree insertion algorithm. This is called the *Baxter P-symbol*
in [Gir12]_ Definition 4.1.
.. NOTE::
This method only works when ``p`` is a permutation. For words
with repeated letters, it would return two "right binary
search trees" (in the terminology of [Gir12]_), which conflicts
with the definition in [Gir12]_.
EXAMPLES::
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([]))
(., .)
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([1, 2, 3]))
(1[., 2[., 3[., .]]], 3[2[1[., .], .], .])
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([3, 4, 1, 2]))
(3[1[., 2[., .]], 4[., .]], 2[1[., .], 4[3[., .], .]])
"""
from sage.combinat.binary_tree import LabelledBinaryTree
left = LabelledBinaryTree(None)
right = LabelledBinaryTree(None)
for a in p:
left = left.binary_search_insert(a)
for a in reversed(p):
right = right.binary_search_insert(a)
return (left, right)
def to_pair_of_twin_binary_trees(self, p):
r"""
Apply a bijection between Baxter permutations of size ``self._n``
and the set of pairs of twin binary trees with ``self._n`` nodes.
INPUT:
- ``p`` -- a Baxter permutation.
OUTPUT:
The pair of twin binary trees `(T_L, T_R)` where `T_L`
(resp. `T_R`) is obtained by inserting the letters of ``p`` from
left to right (resp. right to left) following the binary search
tree insertion algorithm. This is called the *Baxter P-symbol*
in [Gir2012]_ Definition 4.1.
.. NOTE::
This method only works when ``p`` is a permutation. For words
with repeated letters, it would return two "right binary
search trees" (in the terminology of [Gir2012]_), which conflicts
with the definition in [Gir2012]_.
EXAMPLES::
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([]))
(., .)
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([1, 2, 3]))
(1[., 2[., 3[., .]]], 3[2[1[., .], .], .])
sage: BaxterPermutations().to_pair_of_twin_binary_trees(Permutation([3, 4, 1, 2]))
(3[1[., 2[., .]], 4[., .]], 2[1[., .], 4[3[., .], .]])
"""
from sage.combinat.binary_tree import LabelledBinaryTree
left = LabelledBinaryTree(None)
right = LabelledBinaryTree(None)
for a in p:
left = left.binary_search_insert(a)
for a in reversed(p):
right = right.binary_search_insert(a)
return (left, right)