# Tamari Interval-posets¶

This module implements Tamari interval-posets: combinatorial objects which represent intervals of the Tamari order. They have been introduced in [CP2015] and allow for many combinatorial operations on Tamari intervals. In particular, they are linked to DyckWords and BinaryTrees. An introduction into Tamari interval-posets is given in Chapter 7 of [Pons2013].

The Tamari lattice can be defined as a lattice structure on either of several classes of Catalan objects, especially binary trees and Dyck paths [Tam1962] [HT1972] [Sta-EC2]. An interval can be seen as a pair of comparable elements. The number of intervals has been given in [Cha2008].

AUTHORS:

• Viviane Pons 2014: initial implementation
• Frédéric Chapoton 2014: review
• Darij Grinberg 2014: review
• Travis Scrimshaw 2014: review
class sage.combinat.interval_posets.TamariIntervalPoset(parent, size, relations=[], check=True)

The class of Tamari interval-posets.

An interval-poset is a labelled poset of size $$n$$, with labels $$1, 2, \ldots, n$$, satisfying the following conditions:

• if $$a < c$$ (as integers) and $$a$$ precedes $$c$$ in the poset, then, for all $$b$$ such that $$a < b < c$$, $$b$$ precedes $$c$$,
• if $$a < c$$ (as integers) and $$c$$ precedes $$a$$ in the poset, then, for all $$b$$ such that $$a < b < c$$, $$b$$ precedes $$a$$.

We use the word “precedes” here to distinguish the poset order and the natural order on numbers. “Precedes” means “is smaller than with respect to the poset structure”; this does not imply a covering relation.

Interval-posets of size $$n$$ are in bijection with intervals of the Tamari lattice of binary trees of size $$n$$. Specifically, if $$P$$ is an interval-poset of size $$n$$, then the set of linear extensions of $$P$$ (as permutations in $$S_n$$) is an interval in the right weak order (see permutohedron_lequal()), and is in fact the preimage of an interval in the Tamari lattice (of binary trees of size $$n$$) under the operation which sends a permutation to its right-to-left binary search tree (binary_search_tree() with the left_to_right variable set to False) without its labelling.

INPUT:

• size – an integer, the size of the interval-posets (number of vertices)
• relations – a list (or tuple) of pairs (a,b) (themselves lists or tuples), each representing a relation of the form ‘$$a$$ precedes $$b$$’ in the poset.
• check – (default: True) whether to check the interval-poset condition or not.

Warning

The relations input can be a list or tuple, but not an iterator (nor should its entries be iterators).

NOTATION:

Here and in the following, the signs $$<$$ and $$>$$ always refer to the natural ordering on integers, whereas the word “precedes” refers to the order of the interval-poset. “Minimal” and “maximal” refer to the natural ordering on integers.

The increasing relations of an interval-poset $$P$$ mean the pairs $$(a, b)$$ of elements of $$P$$ such that $$a < b$$ as integers and $$a$$ precedes $$b$$ in $$P$$. The initial forest of $$P$$ is the poset obtained by imposing (only) the increasing relations on the ground set of $$P$$. It is a sub-interval poset of $$P$$, and is a forest with its roots on top. This forest is usually given the structure of a planar forest by ordering brother nodes by their labels; it then has the property that if its nodes are traversed in post-order (see post_order_traversal(), and traverse the trees of the forest from left to right as well), then the labels encountered are $$1, 2, \ldots, n$$ in this order.

The decreasing relations of an interval-poset $$P$$ mean the pairs $$(a, b)$$ of elements of $$P$$ such that $$b < a$$ as integers and $$a$$ precedes $$b$$ in $$P$$. The final forest of $$P$$ is the poset obtained by imposing (only) the decreasing relations on the ground set of $$P$$. It is a sub-interval poset of $$P$$, and is a forest with its roots on top. This forest is usually given the structure of a planar forest by ordering brother nodes by their labels; it then has the property that if its nodes are traversed in pre-order (see pre_order_traversal(), and traverse the trees of the forest from left to right as well), then the labels encountered are $$1, 2, \ldots, n$$ in this order.

EXAMPLES:

sage: TamariIntervalPoset(0,[])
The Tamari interval of size 0 induced by relations []
sage: TamariIntervalPoset(3,[])
The Tamari interval of size 3 induced by relations []
sage: TamariIntervalPoset(3,[(1,2)])
The Tamari interval of size 3 induced by relations [(1, 2)]
sage: TamariIntervalPoset(3,[(1,2),(2,3)])
The Tamari interval of size 3 induced by relations [(1, 2), (2, 3)]
sage: TamariIntervalPoset(3,[(1,2),(2,3),(1,3)])
The Tamari interval of size 3 induced by relations [(1, 2), (2, 3)]
sage: TamariIntervalPoset(3,[(1,2),(3,2)])
The Tamari interval of size 3 induced by relations [(1, 2), (3, 2)]
sage: TamariIntervalPoset(3,[[1,2],[2,3]])
The Tamari interval of size 3 induced by relations [(1, 2), (2, 3)]
sage: TamariIntervalPoset(3,[[1,2],[2,3],[1,2],[1,3]])
The Tamari interval of size 3 induced by relations [(1, 2), (2, 3)]

sage: TamariIntervalPoset(3,[(3,4)])
Traceback (most recent call last):
...
ValueError: the relations do not correspond to the size of the poset

sage: TamariIntervalPoset(2,[(2,1),(1,2)])
Traceback (most recent call last):
...
ValueError: The graph is not directed acyclic

sage: TamariIntervalPoset(3,[(1,3)])
Traceback (most recent call last):
...
ValueError: this does not satisfy the Tamari interval-poset condition


It is also possible to transform a poset directly into an interval-poset:

sage: TIP = TamariIntervalPosets()
sage: p = Poset(([1,2,3], [(1,2)]))
sage: TIP(p)
The Tamari interval of size 3 induced by relations [(1, 2)]
sage: TIP(Poset({1: []}))
The Tamari interval of size 1 induced by relations []
sage: TIP(Poset({}))
The Tamari interval of size 0 induced by relations []

binary_trees()

Return an iterator on all the binary trees in the interval represented by self.

EXAMPLES:

sage: list(TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)]).binary_trees())
[[., [[., [., .]], .]],
[[., [., [., .]]], .],
[., [[[., .], .], .]],
[[., [[., .], .]], .]]
sage: set(TamariIntervalPoset(4,[]).binary_trees()) == set(BinaryTrees(4))
True

complement()

Return the complement of the interval-poset self.

If $$P$$ is a Tamari interval-poset of size $$n$$, then the complement of $$P$$ is defined as the interval-poset $$Q$$ whose base set is $$[n] = \{1, 2, \ldots, n\}$$ (just as for $$P$$), but whose order relation has $$a$$ precede $$b$$ if and only if $$n + 1 - a$$ precedes $$n + 1 - b$$ in $$P$$.

In terms of the Tamari lattice, the complement is the symmetric of self. It is formed from the left-right symmeterized of the binary trees of the interval (switching left and right subtrees, see left_right_symmetry()). In particular, initial intervals are sent to final intervals and vice-versa.

EXAMPLES:

sage: TamariIntervalPoset(3, [(2, 1), (3, 1)]).complement()
The Tamari interval of size 3 induced by relations [(1, 3), (2, 3)]
sage: TamariIntervalPoset(0, []).complement()
The Tamari interval of size 0 induced by relations []
sage: ip = TamariIntervalPoset(4, [(1, 2), (2, 4), (3, 4)])
sage: ip.complement() == TamariIntervalPoset(4, [(2, 1), (3, 1), (4, 3)])
True
sage: ip.lower_binary_tree() == ip.complement().upper_binary_tree().left_right_symmetry()
True
sage: ip.upper_binary_tree() == ip.complement().lower_binary_tree().left_right_symmetry()
True
sage: ip.is_initial_interval()
True
sage: ip.complement().is_final_interval()
True

contains_binary_tree(binary_tree)

Return whether the interval represented by self contains the binary tree binary_tree.

INPUT:

• binary_tree – a binary tree

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.contains_binary_tree(BinaryTree([[None,[None,[]]],None]))
True
sage: ip.contains_binary_tree(BinaryTree([None,[[[],None],None]]))
True
sage: ip.contains_binary_tree(BinaryTree([[],[[],None]]))
False
sage: ip.contains_binary_tree(ip.lower_binary_tree())
True
sage: ip.contains_binary_tree(ip.upper_binary_tree())
True
sage: all(ip.contains_binary_tree(bt) for bt in ip.binary_trees())
True

contains_dyck_word(dyck_word)

Return whether the interval represented by self contains the Dyck word dyck_word.

INPUT:

• dyck_word – a Dyck word

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.contains_dyck_word(DyckWord([1,1,1,0,0,0,1,0]))
True
sage: ip.contains_dyck_word(DyckWord([1,1,0,1,0,1,0,0]))
True
sage: ip.contains_dyck_word(DyckWord([1,0,1,1,0,1,0,0]))
False
sage: ip.contains_dyck_word(ip.lower_dyck_word())
True
sage: ip.contains_dyck_word(ip.upper_dyck_word())
True
sage: all(ip.contains_dyck_word(bt) for bt in ip.dyck_words())
True

contains_interval(other)

Return whether the interval represented by other is contained in self as an interval of the Tamari lattice.

In terms of interval-posets, it means that all relations of self are relations of other.

INPUT:

• other – an interval-poset

EXAMPLES:

sage: ip1 = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip2 = TamariIntervalPoset(4,[(2,3)])
sage: ip2.contains_interval(ip1)
True
sage: ip3 = TamariIntervalPoset(4,[(2,1)])
sage: ip2.contains_interval(ip3)
False
sage: ip4 = TamariIntervalPoset(3,[(2,3)])
sage: ip2.contains_interval(ip4)
False

cubical_coordinates()

Return the cubical coordinates of self.

This provides a fast and natural way to order the set of interval-posets of a given size.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.cubical_coordinates()
(-1, -2, 0)


REFERENCES:

decomposition_to_triple()

Decompose an interval-poset into a triple (left, right, r).

For the inverse method, see TamariIntervalPosets.recomposition_from_triple().

OUTPUT:

a triple (left, right, r) where left and right are interval-posets and r (an integer) is the parameter of the decomposition.

EXAMPLES:

sage: tip = TamariIntervalPoset(8, [(1,2), (2,4), (3,4), (6,7), (3,2), (5,4), (6,4), (8,7)])
sage: tip.decomposition_to_triple()
(The Tamari interval of size 3 induced by relations [(1, 2), (3, 2)],
The Tamari interval of size 4 induced by relations [(2, 3), (4, 3)],
2)
sage: tip == TamariIntervalPosets.recomposition_from_triple(*tip.decomposition_to_triple())
True


REFERENCES:

decreasing_children(v)

Return the children of v in the final forest of self.

INPUT:

• v – an integer representing a vertex of self (between 1 and size)

OUTPUT:

The list of all children of v in the final forest of self, in increasing order.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.decreasing_children(2)
[3, 5]
sage: ip.decreasing_children(3)
[4]
sage: ip.decreasing_children(1)
[]

decreasing_cover_relations()

Return the cover relations of the final forest of self.

This is the poset formed by keeping only the relations of the form $$a$$ precedes $$b$$ with $$a > b$$.

The final forest of self is a forest with its roots being on top. It is also called the decreasing poset of self.

Warning

This method computes the cover relations of the final forest. This is not identical with the cover relations of self which happen to be decreasing!

EXAMPLES:

sage: TamariIntervalPoset(4,[(2,1),(3,2),(3,4),(4,2)]).decreasing_cover_relations()
[(4, 2), (3, 2), (2, 1)]
sage: TamariIntervalPoset(4,[(2,1),(4,3),(2,3)]).decreasing_cover_relations()
[(4, 3), (2, 1)]
sage: TamariIntervalPoset(3,[(2,1),(3,1),(3,2)]).decreasing_cover_relations()
[(3, 2), (2, 1)]

decreasing_parent(v)

Return the vertex parent of v in the final forest of self.

This is the highest (as integer!) vertex $$a < v$$ such that v precedes a. If there is no such vertex (that is, $$v$$ is a decreasing root), then None is returned.

INPUT:

• v – an integer representing a vertex of self (between 1 and size)

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.decreasing_parent(4)
3
sage: ip.decreasing_parent(3)
2
sage: ip.decreasing_parent(5)
2
sage: ip.decreasing_parent(2) is None
True

decreasing_roots()

Return the root vertices of the final forest of self.

These are the vertices $$b$$ such that there is no $$a < b$$ with $$b$$ preceding $$a$$.

OUTPUT:

The list of all roots of the final forest of self, in increasing order.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.decreasing_roots()
[1, 2]
sage: ip.final_forest().decreasing_roots()
[1, 2]

dyck_words()

Return an iterator on all the Dyck words in the interval represented by self.

EXAMPLES:

sage: list(TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)]).dyck_words())
[[1, 1, 1, 0, 0, 1, 0, 0],
[1, 1, 1, 0, 0, 0, 1, 0],
[1, 1, 0, 1, 0, 1, 0, 0],
[1, 1, 0, 1, 0, 0, 1, 0]]
sage: set(TamariIntervalPoset(4,[]).dyck_words()) == set(DyckWords(4))
True

final_forest()

Return the final forest of self, i.e., the interval-poset formed with only the decreasing relations of self.

EXAMPLES:

sage: TamariIntervalPoset(4,[(2,1),(3,2),(3,4),(4,2)]).final_forest()
The Tamari interval of size 4 induced by relations [(4, 2), (3, 2), (2, 1)]
sage: ip = TamariIntervalPoset(3,[(2,1),(3,1)])
sage: ip.final_forest() == ip
True

ge(e1, e2)

Return whether e2 precedes or equals e1 in self.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.ge(2,1)
True
sage: ip.ge(3,1)
True
sage: ip.ge(3,2)
True
sage: ip.ge(4,3)
False
sage: ip.ge(1,1)
True

grafting_tree()

Return the grafting tree of the interval-poset.

For the inverse method, see TamariIntervalPosets.from_grafting_tree().

EXAMPLES:

sage: tip = TamariIntervalPoset(8, [(1,2), (2,4), (3,4), (6,7), (3,2), (5,4), (6,4), (8,7)])
sage: tip.grafting_tree()
2[1[0[., .], 0[., .]], 0[., 1[0[., .], 0[., .]]]]
sage: tip == TamariIntervalPosets.from_grafting_tree(tip.grafting_tree())
True


REFERENCES:

gt(e1, e2)

Return whether e2 strictly precedes e1 in self.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.gt(2,1)
True
sage: ip.gt(3,1)
True
sage: ip.gt(3,2)
True
sage: ip.gt(4,3)
False
sage: ip.gt(1,1)
False

increasing_children(v)

Return the children of v in the initial forest of self.

INPUT:

• v – an integer representing a vertex of self (between 1 and size)

OUTPUT:

The list of all children of v in the initial forest of self, in decreasing order.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.increasing_children(2)
[1]
sage: ip.increasing_children(5)
[4, 3]
sage: ip.increasing_children(1)
[]

increasing_cover_relations()

Return the cover relations of the initial forest of self.

This is the poset formed by keeping only the relations of the form $$a$$ precedes $$b$$ with $$a < b$$.

The initial forest of self is a forest with its roots being on top. It is also called the increasing poset of self.

Warning

This method computes the cover relations of the initial forest. This is not identical with the cover relations of self which happen to be increasing!

EXAMPLES:

sage: TamariIntervalPoset(4,[(1,2),(3,2),(2,4),(3,4)]).increasing_cover_relations()
[(1, 2), (2, 4), (3, 4)]
sage: TamariIntervalPoset(3,[(1,2),(1,3),(2,3)]).increasing_cover_relations()
[(1, 2), (2, 3)]

increasing_parent(v)

Return the vertex parent of v in the initial forest of self.

This is the lowest (as integer!) vertex $$b > v$$ such that $$v$$ precedes $$b$$. If there is no such vertex (that is, $$v$$ is an increasing root), then None is returned.

INPUT:

• v – an integer representing a vertex of self (between 1 and size)

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.increasing_parent(1)
2
sage: ip.increasing_parent(3)
5
sage: ip.increasing_parent(4)
5
sage: ip.increasing_parent(5) is None
True

increasing_roots()

Return the root vertices of the initial forest of self.

These are the vertices $$a$$ of self such that there is no $$b > a$$ with $$a$$ precedes $$b$$.

OUTPUT:

The list of all roots of the initial forest of self, in decreasing order.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.increasing_roots()
[6, 5, 2]
sage: ip.initial_forest().increasing_roots()
[6, 5, 2]

initial_forest()

Return the initial forest of self, i.e., the interval-poset formed from only the increasing relations of self.

EXAMPLES:

sage: TamariIntervalPoset(4,[(1,2),(3,2),(2,4),(3,4)]).initial_forest()
The Tamari interval of size 4 induced by relations [(1, 2), (2, 4), (3, 4)]
sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.initial_forest() == ip
True

insertion(i)

Return the Tamari insertion of an integer $$i$$ into the interval-poset self.

If $$P$$ is a Tamari interval-poset of size $$n$$ and $$i$$ is an integer with $$1 \leq i \leq n+1$$, then the Tamari insertion of $$i$$ into $$P$$ is defined as the Tamari interval-poset of size $$n+1$$ which corresponds to the interval $$[C_1, C_2]$$ on the Tamari lattice, where the binary trees $$C_1$$ and $$C_2$$ are defined as follows: We write the interval-poset $$P$$ as $$[B_1, B_2]$$ for two binary trees $$B_1$$ and $$B_2$$. We label the vertices of each of these two trees with the integers $$1, 2, \ldots, i-1, i+1, i+2, \ldots, n+1$$ in such a way that the trees are binary search trees (this labelling is unique). Then, we insert $$i$$ into each of these trees (in the way as explained in binary_search_insert()). The shapes of the resulting two trees are denoted $$C_1$$ and $$C_2$$.

An alternative way to construct the insertion of $$i$$ into $$P$$ is by relabeling each vertex $$u$$ of $$P$$ satisfying $$u \geq i$$ (as integers) as $$u+1$$, and then adding a vertex $$i$$ which should precede $$i-1$$ and $$i+1$$.

Todo

To study this, it would be more natural to define interval-posets on arbitrary ordered sets rather than just on $$\{1, 2, \ldots, n\}$$.

EXAMPLES:

sage: ip = TamariIntervalPoset(4, [(2, 3), (4, 3)]); ip
The Tamari interval of size 4 induced by relations [(2, 3), (4, 3)]
sage: ip.insertion(1)
The Tamari interval of size 5 induced by relations [(1, 2), (3, 4), (5, 4)]
sage: ip.insertion(2)
The Tamari interval of size 5 induced by relations [(2, 3), (3, 4), (5, 4), (2, 1)]
sage: ip.insertion(3)
The Tamari interval of size 5 induced by relations [(2, 4), (3, 4), (5, 4), (3, 2)]
sage: ip.insertion(4)
The Tamari interval of size 5 induced by relations [(2, 3), (4, 5), (5, 3), (4, 3)]
sage: ip.insertion(5)
The Tamari interval of size 5 induced by relations [(2, 3), (5, 4), (4, 3)]

sage: ip = TamariIntervalPoset(0, [])
sage: ip.insertion(1)
The Tamari interval of size 1 induced by relations []

sage: ip = TamariIntervalPoset(1, [])
sage: ip.insertion(1)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: ip.insertion(2)
The Tamari interval of size 2 induced by relations [(2, 1)]

intersection(other)

Return the interval-poset formed by combining the relations from both self and other. It corresponds to the intersection of the two corresponding intervals of the Tamari lattice.

INPUT:

• other – an interval-poset of the same size as self

EXAMPLES:

sage: ip1 = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip2 = TamariIntervalPoset(4,[(4,3)])
sage: ip1.intersection(ip2)
The Tamari interval of size 4 induced by relations [(1, 2), (2, 3), (4, 3)]
sage: ip3 = TamariIntervalPoset(4,[(2,1)])
sage: ip1.intersection(ip3)
Traceback (most recent call last):
...
ValueError: this intersection is empty, it does not correspond to an interval-poset
sage: ip4 = TamariIntervalPoset(3,[(2,3)])
sage: ip2.intersection(ip4)
Traceback (most recent call last):
...
ValueError: intersections are only possible on interval-posets of the same size

interval_cardinality()

Return the cardinality of the interval, i.e., the number of elements (binary trees or Dyck words) in the interval represented by self.

Not to be confused with size() which is the number of vertices.

EXAMPLES:

sage: TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)]).interval_cardinality()
4
sage: TamariIntervalPoset(4,[]).interval_cardinality()
14
sage: TamariIntervalPoset(4,[(1,2),(2,3),(3,4)]).interval_cardinality()
1

is_connected()

Return whether self is a connected Tamari interval.

This means that the Hasse diagram is connected.

This condition is invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3) if T.is_connected()])
8

is_dexter()

Return whether self is a dexter Tamari interval.

This is defined by an exclusion pattern in the Hasse diagram. See the code for the exact description.

This condition is not invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3) if T.is_dexter()])
12

is_exceptional()

Return whether self is an exceptional Tamari interval.

This is defined by exclusion of a simple pattern in the Hasse diagram, namely there is no configuration y <-- x --> z with $$1 \leq y < x < z \leq n$$.

This condition is invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3)
....:     if T.is_exceptional()])
12

is_final_interval()

Return if self corresponds to a final interval of the Tamari lattice.

This means that its upper end is the largest element of the lattice. It consists of checking that self does not contain any increasing relations.

EXAMPLES:

sage: ip = TamariIntervalPoset(4, [(4, 3), (3, 1), (2, 1)])
sage: ip.is_final_interval()
True
sage: ip.upper_dyck_word()
[1, 1, 1, 1, 0, 0, 0, 0]
sage: ip = TamariIntervalPoset(4, [(4, 3), (3, 1), (2, 1), (2, 3)])
sage: ip.is_final_interval()
False
sage: ip.upper_dyck_word()
[1, 1, 0, 1, 1, 0, 0, 0]
sage: all(dw.tamari_interval(DyckWord([1, 1, 1, 0, 0, 0])).is_final_interval() for dw in DyckWords(3))
True

is_indecomposable()

Return whether self is an indecomposable Tamari interval.

This is the terminology of [Cha2008].

This means that the upper binary tree has an empty left subtree.

This condition is not invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3)
....:      if T.is_indecomposable()])
8

is_infinitely_modern()

Return whether self is an infinitely-modern Tamari interval.

This is defined by the exclusion of the configuration $$i \rightarrow i + 1$$ and $$j + 1 \rightarrow j$$ with $$i < j$$.

This condition is invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3)
....:     if T.is_infinitely_modern()])
12


REFERENCES:

is_initial_interval()

Return if self corresponds to an initial interval of the Tamari lattice.

This means that its lower end is the smallest element of the lattice. It consists of checking that self does not contain any decreasing relations.

EXAMPLES:

sage: ip = TamariIntervalPoset(4, [(1, 2), (2, 4), (3, 4)])
sage: ip.is_initial_interval()
True
sage: ip.lower_dyck_word()
[1, 0, 1, 0, 1, 0, 1, 0]
sage: ip = TamariIntervalPoset(4, [(1, 2), (2, 4), (3, 4), (3, 2)])
sage: ip.is_initial_interval()
False
sage: ip.lower_dyck_word()
[1, 0, 1, 1, 0, 0, 1, 0]
sage: all(DyckWord([1,0,1,0,1,0]).tamari_interval(dw).is_initial_interval() for dw in DyckWords(3))
True

is_linear_extension(perm)

Return whether the permutation perm is a linear extension of self.

INPUT:

• perm – a permutation of the size of self

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip.is_linear_extension([1,4,2,3])
True
sage: ip.is_linear_extension(Permutation([1,4,2,3]))
True
sage: ip.is_linear_extension(Permutation([1,4,3,2]))
False

is_modern()

Return whether self is a modern Tamari interval.

This is defined by exclusion of a simple pattern in the Hasse diagram, namely there is no configuration $$y \rightarrow x \leftarrow z$$ with $$1 \leq y < x < z \leq n$$.

This condition is invariant under complementation.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3) if T.is_modern()])
12


REFERENCES:

is_new()

Return whether self is a new Tamari interval.

Here ‘new’ means that the interval is not contained in any facet of the associahedron. This condition is invariant under complementation.

They have been considered in section 9 of [Cha2008].

EXAMPLES:

sage: TIP4 = TamariIntervalPosets(4)
sage: len([u for u in TIP4 if u.is_new()])
12

sage: TIP3 = TamariIntervalPosets(3)
sage: len([u for u in TIP3 if u.is_new()])
3

is_simple()

Return whether self is a simple Tamari interval.

Here ‘simple’ means that the interval contains a unique binary tree.

These intervals define the simple modules over the incidence algebras of the Tamari lattices.

EXAMPLES:

sage: TIP4 = TamariIntervalPosets(4)
sage: len([u for u in TIP4 if u.is_simple()])
14

sage: TIP3 = TamariIntervalPosets(3)
sage: len([u for u in TIP3 if u.is_simple()])
5

is_synchronized()

Return whether self is a synchronized Tamari interval.

This means that the upper and lower binary trees have the same canopee. This condition is invariant under complementation.

This has been considered in [FPR2015]. The numbers of synchronized intervals are given by the sequence OEIS sequence A000139.

EXAMPLES:

sage: len([T for T in TamariIntervalPosets(3)
....:     if T.is_synchronized()])
6

latex_options()

Return the latex options for use in the _latex_ function as a dictionary.

The default values are set using the options.

• tikz_scale – (default: 1) scale for use with the tikz package
• line_width – (default: 1) value representing the line width (additionally scaled by tikz_scale)
• color_decreasing – (default: 'red') the color for decreasing relations
• color_increasing – (default: 'blue') the color for increasing relations
• hspace – (default: 1) the difference between horizontal coordinates of adjacent vertices
• vspace – (default: 1) the difference between vertical coordinates of adjacent vertices

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.latex_options()['color_decreasing']
'red'
sage: ip.latex_options()['hspace']
1

le(e1, e2)

Return whether e1 precedes or equals e2 in self.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.le(1,2)
True
sage: ip.le(1,3)
True
sage: ip.le(2,3)
True
sage: ip.le(3,4)
False
sage: ip.le(1,1)
True

left_branch_involution()

Return the image of self by the left-branch involution.

OUTPUT: an interval-poset

EXAMPLES:

sage: tip = TamariIntervalPoset(8, [(1,2), (2,4), (3,4), (6,7), (3,2), (5,4), (6,4), (8,7)])
sage: t = tip.left_branch_involution(); t
The Tamari interval of size 8 induced by relations [(1, 6), (2, 6),
(3, 5), (4, 5), (5, 6), (6, 8), (7, 8), (7, 6), (4, 3), (3, 1),
(2, 1)]
sage: t.left_branch_involution() == tip
True


REFERENCES:

linear_extensions()

Return an iterator on the permutations which are linear extensions of self.

They form an interval of the right weak order (also called the right permutohedron order – see permutohedron_lequal() for a definition).

EXAMPLES:

sage: ip = TamariIntervalPoset(3,[(1,2),(3,2)])
sage: list(ip.linear_extensions())
[[3, 1, 2], [1, 3, 2]]
sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: list(ip.linear_extensions())
[[4, 1, 2, 3], [1, 2, 4, 3], [1, 4, 2, 3]]

lower_binary_tree()

Return the lowest binary tree in the interval of the Tamari lattice represented by self.

This is a binary tree. It is the shape of the unique binary search tree whose left-branch ordered forest (i.e., the result of applying to_ordered_tree_left_branch() and cutting off the root) is the final forest of self.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.lower_binary_tree()
[[., .], [[., [., .]], [., .]]]
sage: TamariIntervalPosets.final_forest(ip.lower_binary_tree()) == ip.final_forest()
True
sage: ip == TamariIntervalPosets.from_binary_trees(ip.lower_binary_tree(),ip.upper_binary_tree())
True

lower_contained_intervals()

If self represents the interval $$[t_1, t_2]$$ of the Tamari lattice, return an iterator on all intervals $$[t_1,t]$$ with $$t \leq t_2$$ for the Tamari lattice.

In terms of interval-posets, it corresponds to adding all possible relations of the form $$n$$ precedes $$m$$ with $$n<m$$.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: list(ip.lower_contained_intervals())
[The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(1, 4), (2, 4), (3, 4), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(2, 3), (3, 4), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(1, 4), (2, 3), (3, 4), (3, 1), (2, 1)]]
sage: ip = TamariIntervalPoset(4,[])
sage: len(list(ip.lower_contained_intervals()))
14

lower_contains_interval(other)

Return whether the interval represented by other is contained in self as an interval of the Tamari lattice and if they share the same lower bound.

As interval-posets, it means that other contains the relations of self plus some extra increasing relations.

INPUT:

• other – an interval-poset

EXAMPLES:

sage: ip1 = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip2 = TamariIntervalPoset(4,[(4,3)])
sage: ip2.lower_contains_interval(ip1)
True
sage: ip2.contains_interval(ip1) and ip2.lower_binary_tree() == ip1.lower_binary_tree()
True
sage: ip3 = TamariIntervalPoset(4,[(4,3),(2,1)])
sage: ip2.contains_interval(ip3)
True
sage: ip2.lower_binary_tree() == ip3.lower_binary_tree()
False
sage: ip2.lower_contains_interval(ip3)
False

lower_dyck_word()

Return the lowest Dyck word in the interval of the Tamari lattice represented by self.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.lower_dyck_word()
[1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0]
sage: TamariIntervalPosets.final_forest(ip.lower_dyck_word()) == ip.final_forest()
True
sage: ip == TamariIntervalPosets.from_dyck_words(ip.lower_dyck_word(),ip.upper_dyck_word())
True

lt(e1, e2)

Return whether e1 strictly precedes e2 in self.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.lt(1,2)
True
sage: ip.lt(1,3)
True
sage: ip.lt(2,3)
True
sage: ip.lt(3,4)
False
sage: ip.lt(1,1)
False

max_linear_extension()

Return the maximal permutation for the right weak order which is a linear extension of self.

This is also the maximal permutation in the sylvester class of self.upper_binary_tree() and is a 132-avoiding permutation.

The right weak order is also known as the right permutohedron order. See permutohedron_lequal() for its definition.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip.max_linear_extension()
[4, 1, 2, 3]
sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.max_linear_extension()
[6, 4, 5, 3, 1, 2]
sage: ip = TamariIntervalPoset(0,[]); ip
The Tamari interval of size 0 induced by relations []
sage: ip.max_linear_extension()
[]
sage: ip = TamariIntervalPoset(5, [(1, 4), (2, 4), (3, 4), (5, 4)]); ip
The Tamari interval of size 5 induced by relations [(1, 4), (2, 4), (3, 4), (5, 4)]
sage: ip.max_linear_extension()
[5, 3, 2, 1, 4]

maximal_chain_binary_trees()

Return an iterator on the binary trees forming a longest chain of self (regarding self as an interval of the Tamari lattice).

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: list(ip.maximal_chain_binary_trees())
[[[., [[., .], .]], .], [., [[[., .], .], .]], [., [[., [., .]], .]]]
sage: ip = TamariIntervalPoset(4,[])
sage: list(ip.maximal_chain_binary_trees())
[[[[[., .], .], .], .],
[[[., [., .]], .], .],
[[., [[., .], .]], .],
[., [[[., .], .], .]],
[., [[., [., .]], .]],
[., [., [[., .], .]]],
[., [., [., [., .]]]]]

maximal_chain_dyck_words()

Return an iterator on the Dyck words forming a longest chain of self (regarding self as an interval of the Tamari lattice).

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: list(ip.maximal_chain_dyck_words())
[[1, 1, 0, 1, 0, 0, 1, 0], [1, 1, 0, 1, 0, 1, 0, 0], [1, 1, 1, 0, 0, 1, 0, 0]]
sage: ip = TamariIntervalPoset(4,[])
sage: list(ip.maximal_chain_dyck_words())
[[1, 0, 1, 0, 1, 0, 1, 0],
[1, 1, 0, 0, 1, 0, 1, 0],
[1, 1, 0, 1, 0, 0, 1, 0],
[1, 1, 0, 1, 0, 1, 0, 0],
[1, 1, 1, 0, 0, 1, 0, 0],
[1, 1, 1, 0, 1, 0, 0, 0],
[1, 1, 1, 1, 0, 0, 0, 0]]

maximal_chain_tamari_intervals()

Return an iterator on the upper contained intervals of one longest chain of the Tamari interval represented by self.

If self represents the interval $$[T_1,T_2]$$ of the Tamari lattice, this returns intervals $$[T',T_2]$$ with $$T'$$ following one longest chain between $$T_1$$ and $$T_2$$.

To obtain a longest chain, we use the Tamari inversions of self. The elements of the chain are obtained by adding one by one the relations $$(b,a)$$ from each Tamari inversion $$(a,b)$$ to self, where the Tamari inversions are taken in lexicographic order.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: list(ip.maximal_chain_tamari_intervals())
[The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (4, 1), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (4, 1), (3, 2), (2, 1)]]
sage: ip = TamariIntervalPoset(4,[])
sage: list(ip.maximal_chain_tamari_intervals())
[The Tamari interval of size 4 induced by relations [],
The Tamari interval of size 4 induced by relations [(2, 1)],
The Tamari interval of size 4 induced by relations [(3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(4, 1), (3, 1), (2, 1)],
The Tamari interval of size 4 induced by relations [(4, 1), (3, 2), (2, 1)],
The Tamari interval of size 4 induced by relations [(4, 2), (3, 2), (2, 1)],
The Tamari interval of size 4 induced by relations [(4, 3), (3, 2), (2, 1)]]

min_linear_extension()

Return the minimal permutation for the right weak order which is a linear extension of self.

This is also the minimal permutation in the sylvester class of self.lower_binary_tree() and is a 312-avoiding permutation.

The right weak order is also known as the right permutohedron order. See permutohedron_lequal() for its definition.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip.min_linear_extension()
[1, 2, 4, 3]
sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)])
sage: ip.min_linear_extension()
[1, 4, 3, 6, 5, 2]
sage: ip = TamariIntervalPoset(0,[])
sage: ip.min_linear_extension()
[]
sage: ip = TamariIntervalPoset(5, [(1, 4), (2, 4), (3, 4), (5, 4)]); ip
The Tamari interval of size 5 induced by relations [(1, 4), (2, 4), (3, 4), (5, 4)]
sage: ip.min_linear_extension()
[1, 2, 3, 5, 4]

new_decomposition()

Return the decomposition of the interval-poset into new interval-posets.

Every interval-poset has a unique decomposition as a planar tree of new interval-posets, as explained in [Cha2008]. This function computes the terms of this decomposition, but not the planar tree.

For the number of terms, you can use instead the method number_of_new_components().

OUTPUT:

a list of new interval-posets.

EXAMPLES:

sage: ex = TamariIntervalPosets(4)[11]
sage: ex.number_of_new_components()
3
sage: ex.new_decomposition()
[The Tamari interval of size 1 induced by relations [],
The Tamari interval of size 2 induced by relations [],
The Tamari interval of size 1 induced by relations []]

number_of_new_components()

Return the number of terms in the decomposition in new interval-posets.

Every interval-poset has a unique decomposition as a planar tree of new interval-posets, as explained in [Cha2008]. This function just computes the number of terms, not the planar tree nor the terms themselves.

EXAMPLES:

sage: TIP4 = TamariIntervalPosets(4)
sage: nb = [u.number_of_new_components() for u in TIP4]
sage: [nb.count(i) for i in range(1, 5)]
[12, 21, 21, 14]

number_of_tamari_inversions()

Return the number of Tamari inversions of self.

This is also the length the longest chain of the Tamari interval represented by self.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.number_of_tamari_inversions()
2
sage: ip = TamariIntervalPoset(4,[])
sage: ip.number_of_tamari_inversions()
6
sage: ip = TamariIntervalPoset(3,[])
sage: ip.number_of_tamari_inversions()
3

plot(**kwds)

Return a picture.

The picture represents the Hasse diagram, where the covers are colored in blue if they are increasing and in red if they are decreasing.

This uses the same coordinates as the latex view.

EXAMPLES:

sage: ti = TamariIntervalPosets(4)[2]
sage: ti.plot()
Graphics object consisting of 6 graphics primitives

poset()

Return self as a labelled poset.

An interval-poset is indeed constructed from a labelled poset which is stored internally. This method allows to access the poset and all the associated methods.

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(3,2),(2,4),(3,4)])
sage: pos = ip.poset(); pos
Finite poset containing 4 elements
sage: pos.maximal_chains()
[[3, 2, 4], [1, 2, 4]]
sage: pos.maximal_elements()
[4]
sage: pos.is_lattice()
False

rise_contact_involution()

Return the image of self by the rise-contact involution.

OUTPUT: an interval-poset

This is defined by conjugating the complement involution by the left-branch involution.

EXAMPLES:

sage: tip = TamariIntervalPoset(8, [(1,2), (2,4), (3,4), (6,7), (3,2), (5,4), (6,4), (8,7)])
sage: t = tip.rise_contact_involution(); t
The Tamari interval of size 8 induced by relations [(2, 8), (3, 8),
(4, 5), (5, 7), (6, 7), (7, 8), (8, 1), (7, 2), (6, 2), (5, 3),
(4, 3), (3, 2), (2, 1)]
sage: t.rise_contact_involution() == tip
True
sage: tip.lower_dyck_word().number_of_touch_points() == t.upper_dyck_word().number_of_initial_rises()
True
sage: tip.number_of_tamari_inversions() == t.number_of_tamari_inversions()
True


REFERENCES:

set_latex_options(D)

Set the latex options for use in the _latex_ function.

The default values are set in the __init__ function.

• tikz_scale – (default: 1) scale for use with the tikz package
• line_width – (default: 1 * tikz_scale) value representing the line width
• color_decreasing – (default: red) the color for decreasing relations
• color_increasing – (default: blue) the color for increasing relations
• hspace – (default: 1) the difference between horizontal coordinates of adjacent vertices
• vspace – (default: 1) the difference between vertical coordinates of adjacent vertices

INPUT:

• D – a dictionary with a list of latex parameters to change

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.latex_options()["color_decreasing"]
'red'
sage: ip.set_latex_options({"color_decreasing":'green'})
sage: ip.latex_options()["color_decreasing"]
'green'
sage: ip.set_latex_options({"color_increasing":'black'})
sage: ip.latex_options()["color_increasing"]
'black'


To change the default options for all interval-posets, use the parent’s latex options:

sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip2 = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip.latex_options()["color_decreasing"]
'red'
sage: ip2.latex_options()["color_decreasing"]
'red'
sage: TamariIntervalPosets.options(latex_color_decreasing='green')
sage: ip.latex_options()["color_decreasing"]
'green'
sage: ip2.latex_options()["color_decreasing"]
'green'


Next we set a local latex option and show the global option does not override it:

sage: ip.set_latex_options({"color_decreasing": 'black'})
sage: ip.latex_options()["color_decreasing"]
'black'
sage: TamariIntervalPosets.options(latex_color_decreasing='blue')
sage: ip.latex_options()["color_decreasing"]
'black'
sage: ip2.latex_options()["color_decreasing"]
'blue'
sage: TamariIntervalPosets.options._reset()

size()

Return the size (number of vertices) of the interval-poset.

EXAMPLES:

sage: TamariIntervalPoset(3,[(2,1),(3,1)]).size()
3

sub_poset(start, end)

Return the renormalized subposet of self consisting solely of integers from start (inclusive) to end (not inclusive).

“Renormalized” means that these integers are relabelled $$1,2,\ldots,k$$ in the obvious way (i.e., by subtracting start - 1).

INPUT:

• start – an integer, the starting vertex (inclusive)
• end – an integer, the ending vertex (not inclusive)

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.subposet(1,3)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: ip.subposet(1,4)
The Tamari interval of size 3 induced by relations [(1, 2), (3, 2)]
sage: ip.subposet(1,5)
The Tamari interval of size 4 induced by relations [(1, 2), (4, 3), (3, 2)]
sage: ip.subposet(1,7) == ip
True
sage: ip.subposet(1,1)
The Tamari interval of size 0 induced by relations []

subposet(start, end)

Return the renormalized subposet of self consisting solely of integers from start (inclusive) to end (not inclusive).

“Renormalized” means that these integers are relabelled $$1,2,\ldots,k$$ in the obvious way (i.e., by subtracting start - 1).

INPUT:

• start – an integer, the starting vertex (inclusive)
• end – an integer, the ending vertex (not inclusive)

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(3,5),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (3, 5), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.subposet(1,3)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: ip.subposet(1,4)
The Tamari interval of size 3 induced by relations [(1, 2), (3, 2)]
sage: ip.subposet(1,5)
The Tamari interval of size 4 induced by relations [(1, 2), (4, 3), (3, 2)]
sage: ip.subposet(1,7) == ip
True
sage: ip.subposet(1,1)
The Tamari interval of size 0 induced by relations []

tamari_inversions()

Return the Tamari inversions of self.

A Tamari inversion is a pair of vertices $$(a,b)$$ with $$a < b$$ such that:

• the decreasing parent of $$b$$ is strictly smaller than $$a$$ (or does not exist), and
• the increasing parent of $$a$$ is strictly bigger than $$b$$ (or does not exist).

“Smaller” and “bigger” refer to the numerical values of the elements, not to the poset order.

This method returns the list of all Tamari inversions in lexicographic order.

The number of Tamari inversions is the length of the longest chain of the Tamari interval represented by self.

Indeed, when an interval consists of just one binary tree, it has no inversion. One can also prove that if a Tamari interval $$I' = [T_1', T_2']$$ is a proper subset of a Tamari interval $$I = [T_1, T_2]$$, then the inversion number of $$I'$$ is strictly lower than the inversion number of $$I$$. And finally, by adding the relation $$(b,a)$$ to the interval-poset where $$(a,b)$$ is the first inversion of $$I$$ in lexicographic order, one reduces the inversion number by exactly $$1$$.

EXAMPLES:

sage: ip = TamariIntervalPoset(3,[])
sage: ip.tamari_inversions()
[(1, 2), (1, 3), (2, 3)]
sage: ip = TamariIntervalPoset(3,[(2,1)])
sage: ip.tamari_inversions()
[(1, 3), (2, 3)]
sage: ip = TamariIntervalPoset(3,[(1,2)])
sage: ip.tamari_inversions()
[(2, 3)]
sage: ip = TamariIntervalPoset(3,[(1,2),(3,2)])
sage: ip.tamari_inversions()
[]
sage: ip = TamariIntervalPoset(4,[(2,4),(3,4),(2,1),(3,1)])
sage: ip.tamari_inversions()
[(1, 4), (2, 3)]
sage: ip = TamariIntervalPoset(4,[])
sage: ip.tamari_inversions()
[(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]
sage: all(len(TamariIntervalPosets.from_binary_trees(bt,bt).tamari_inversions())==0 for bt in BinaryTrees(3))
True
sage: all(len(TamariIntervalPosets.from_binary_trees(bt,bt).tamari_inversions())==0 for bt in BinaryTrees(4))
True

tamari_inversions_iter()

Iterate over the Tamari inversions of self, in lexicographic order.

See tamari_inversions() for the definition of the terms involved.

EXAMPLES:

sage: T = TamariIntervalPoset(5, [[1,2],[3,4],[3,2],[5,2],[4,2]])
sage: list(T.tamari_inversions_iter())
[(4, 5)]

sage: T = TamariIntervalPoset(8, [(2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (8, 7), (6, 4), (5, 4), (4, 3), (3, 2)])
sage: list(T.tamari_inversions_iter())
[(1, 2), (1, 7), (5, 6)]

sage: T = TamariIntervalPoset(1, [])
sage: list(T.tamari_inversions_iter())
[]

sage: T = TamariIntervalPoset(0, [])
sage: list(T.tamari_inversions_iter())
[]

upper_binary_tree()

Return the highest binary tree in the interval of the Tamari lattice represented by self.

This is a binary tree. It is the shape of the unique binary search tree whose right-branch ordered forest (i.e., the result of applying to_ordered_tree_right_branch() and cutting off the root) is the initial forest of self.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.upper_binary_tree()
[[., .], [., [[., .], [., .]]]]
sage: TamariIntervalPosets.initial_forest(ip.upper_binary_tree()) == ip.initial_forest()
True
sage: ip == TamariIntervalPosets.from_binary_trees(ip.lower_binary_tree(),ip.upper_binary_tree())
True

upper_contains_interval(other)

Return whether the interval represented by other is contained in self as an interval of the Tamari lattice and if they share the same upper bound.

As interval-posets, it means that other contains the relations of self plus some extra decreasing relations.

INPUT:

• other – an interval-poset

EXAMPLES:

sage: ip1 = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip2 = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: ip2.upper_contains_interval(ip1)
True
sage: ip2.contains_interval(ip1) and ip2.upper_binary_tree() == ip1.upper_binary_tree()
True
sage: ip3 = TamariIntervalPoset(4,[(1,2),(2,3),(3,4)])
sage: ip2.upper_contains_interval(ip3)
False
sage: ip2.contains_interval(ip3)
True
sage: ip2.upper_binary_tree() == ip3.upper_binary_tree()
False

upper_dyck_word()

Return the highest Dyck word in the interval of the Tamari lattice represented by self.

EXAMPLES:

sage: ip = TamariIntervalPoset(6,[(3,2),(4,3),(5,2),(6,5),(1,2),(4,5)]); ip
The Tamari interval of size 6 induced by relations [(1, 2), (4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]
sage: ip.upper_dyck_word()
[1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0]
sage: TamariIntervalPosets.initial_forest(ip.upper_dyck_word()) == ip.initial_forest()
True
sage: ip == TamariIntervalPosets.from_dyck_words(ip.lower_dyck_word(),ip.upper_dyck_word())
True

class sage.combinat.interval_posets.TamariIntervalPosets

Factory for interval-posets.

INPUT:

• size – (optional) an integer

OUTPUT:

• the set of all interval-posets (of the given size if specified)

EXAMPLES:

sage: TamariIntervalPosets()
Interval-posets

sage: TamariIntervalPosets(2)
Interval-posets of size 2


Note

This is a factory class whose constructor returns instances of subclasses.

static check_poset(poset)

Check if the given poset poset is a interval-poset, that is, if it satisfies the following properties:

• Its labels are exactly $$1, \ldots, n$$ where $$n$$ is its size.
• If $$a < c$$ (as numbers) and $$a$$ precedes $$c$$, then $$b$$ precedes $$c$$ for all $$b$$ such that $$a < b < c$$.
• If $$a < c$$ (as numbers) and $$c$$ precedes $$a$$, then $$b$$ precedes $$a$$ for all $$b$$ such that $$a < b < c$$.

INPUT:

• poset – a finite labeled poset

EXAMPLES:

sage: p = Poset(([1,2,3],[(1,2),(3,2)]))
sage: TamariIntervalPosets.check_poset(p)
True
sage: p = Poset(([2,3],[(3,2)]))
sage: TamariIntervalPosets.check_poset(p)
False
sage: p = Poset(([1,2,3],[(3,1)]))
sage: TamariIntervalPosets.check_poset(p)
False
sage: p = Poset(([1,2,3],[(1,3)]))
sage: TamariIntervalPosets.check_poset(p)
False

static final_forest(element)

Return the final forest of a binary tree, an interval-poset or a Dyck word.

A final forest is an interval-poset corresponding to a final interval of the Tamari lattice, i.e., containing only decreasing relations.

It can be constructed from a binary tree by its binary search tree labeling with the rule: $$b$$ precedes $$a$$ in the final forest iff $$b$$ is in the right subtree of $$a$$ in the binary search tree.

INPUT:

• element – a binary tree, a Dyck word or an interval-poset

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: TamariIntervalPosets.final_forest(ip)
The Tamari interval of size 4 induced by relations [(4, 3)]


From binary trees:

sage: bt = BinaryTree(); bt
.
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 0 induced by relations []
sage: bt = BinaryTree([]); bt
[., .]
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 1 induced by relations []
sage: bt = BinaryTree([[],None]); bt
[[., .], .]
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 2 induced by relations []
sage: bt = BinaryTree([None,[]]); bt
[., [., .]]
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 2 induced by relations [(2, 1)]
sage: bt = BinaryTree([[],[]]); bt
[[., .], [., .]]
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 3 induced by relations [(3, 2)]
sage: bt = BinaryTree([[None,[[],None]],[]]); bt
[[., [[., .], .]], [., .]]
sage: TamariIntervalPosets.final_forest(bt)
The Tamari interval of size 5 induced by relations [(5, 4), (3, 1), (2, 1)]


From Dyck words:

sage: dw = DyckWord([1,0])
sage: TamariIntervalPosets.final_forest(dw)
The Tamari interval of size 1 induced by relations []
sage: dw = DyckWord([1,1,0,1,0,0,1,1,0,0])
sage: TamariIntervalPosets.final_forest(dw)
The Tamari interval of size 5 induced by relations [(5, 4), (3, 1), (2, 1)]

static from_binary_trees(tree1, tree2)

Return the interval-poset corresponding to the interval [tree1, tree2] of the Tamari lattice.

Raise an exception if tree1 is not $$\leq$$ tree2 in the Tamari lattice.

INPUT:

• tree1 – a binary tree
• tree2 – a binary tree greater or equal than tree1 for the Tamari lattice

EXAMPLES:

sage: tree1 = BinaryTree([[],None])
sage: tree2 = BinaryTree([None,[]])
sage: TamariIntervalPosets.from_binary_trees(tree1,tree2)
The Tamari interval of size 2 induced by relations []
sage: TamariIntervalPosets.from_binary_trees(tree1,tree1)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: TamariIntervalPosets.from_binary_trees(tree2,tree2)
The Tamari interval of size 2 induced by relations [(2, 1)]

sage: tree1 = BinaryTree([[],[[None,[]],[]]])
sage: tree2 = BinaryTree([None,[None,[None,[[],[]]]]])
sage: TamariIntervalPosets.from_binary_trees(tree1,tree2)
The Tamari interval of size 6 induced by relations [(4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]

sage: tree3 = BinaryTree([None,[None,[[],[None,[]]]]])
sage: TamariIntervalPosets.from_binary_trees(tree1,tree3)
Traceback (most recent call last):
...
ValueError: the two binary trees are not comparable on the Tamari lattice
sage: TamariIntervalPosets.from_binary_trees(tree1,BinaryTree())
Traceback (most recent call last):
...
ValueError: the two binary trees are not comparable on the Tamari lattice

static from_dyck_words(dw1, dw2)

Return the interval-poset corresponding to the interval [dw1, dw2] of the Tamari lattice.

Raise an exception if the two Dyck words dw1 and dw2 do not satisfy dw1 $$\leq$$ dw2 in the Tamari lattice.

INPUT:

• dw1 – a Dyck word
• dw2 – a Dyck word greater or equal than dw1 for the Tamari lattice

EXAMPLES:

sage: dw1 = DyckWord([1,0,1,0])
sage: dw2 = DyckWord([1,1,0,0])
sage: TamariIntervalPosets.from_dyck_words(dw1,dw2)
The Tamari interval of size 2 induced by relations []
sage: TamariIntervalPosets.from_dyck_words(dw1,dw1)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: TamariIntervalPosets.from_dyck_words(dw2,dw2)
The Tamari interval of size 2 induced by relations [(2, 1)]

sage: dw1 = DyckWord([1,0,1,1,1,0,0,1,1,0,0,0])
sage: dw2 = DyckWord([1,1,1,1,0,1,1,0,0,0,0,0])
sage: TamariIntervalPosets.from_dyck_words(dw1,dw2)
The Tamari interval of size 6 induced by relations [(4, 5), (6, 5), (5, 2), (4, 3), (3, 2)]

sage: dw3 = DyckWord([1,1,1,0,1,1,1,0,0,0,0,0])
sage: TamariIntervalPosets.from_dyck_words(dw1,dw3)
Traceback (most recent call last):
...
ValueError: the two Dyck words are not comparable on the Tamari lattice
sage: TamariIntervalPosets.from_dyck_words(dw1,DyckWord([1,0]))
Traceback (most recent call last):
...
ValueError: the two Dyck words are not comparable on the Tamari lattice

static from_grafting_tree(tree)

Return an interval-poset from a grafting tree.

For the inverse method, see TamariIntervalPoset.grafting_tree().

EXAMPLES:

sage: tip = TamariIntervalPoset(8, [(1,2), (2,4), (3,4), (6,7), (3,2), (5,4), (6,4), (8,7)])
sage: t = tip.grafting_tree()
sage: TamariIntervalPosets.from_grafting_tree(t) == tip
True


REFERENCES:

static from_minimal_schnyder_wood(graph)

Return a Tamari interval built from a minimal Schnyder wood.

This is an implementation of Bernardi and Bonichon’s bijection [BeBo2009].

INPUT:

a minimal Schnyder wood, given as a graph with colored and oriented edges, without the three exterior unoriented edges

The three boundary vertices must be -1, -2 and -3.

One assumes moreover that the embedding around -1 is the list of neighbors of -1 and not just a cyclic permutation of that.

Beware that the embedding convention used here is the opposite of the one used by the plot method.

OUTPUT:

a Tamari interval-poset

EXAMPLES:

A small example:

sage: TIP = TamariIntervalPosets
sage: G = DiGraph([(0,-1,0),(0,-2,1),(0,-3,2)], format='list_of_edges')
sage: G.set_embedding({-1:[0],-2:[0],-3:[0],0:[-1,-2,-3]})
sage: TIP.from_minimal_schnyder_wood(G)
The Tamari interval of size 1 induced by relations []


An example from page 14 of [BeBo2009]:

sage: c0 = [(0,-1),(1,0),(2,0),(4,3),(3,-1),(5,3)]
sage: c1 = [(5,-2),(3,-2),(4,5),(1,3),(2,3),(0,3)]
sage: c2 = [(0,-3),(1,-3),(3,-3),(4,-3),(5,-3),(2,1)]
sage: ed = [(u,v,0) for u,v in c0]
sage: ed += [(u,v,1) for u,v in c1]
sage: ed += [(u,v,2) for u,v in c2]
sage: G = DiGraph(ed, format='list_of_edges')
sage: embed = {-1:[3,0],-2:[5,3],-3:[0,1,3,4,5]}
sage: data_emb = [[3,2,1,-3,-1],[2,3,-3,0],[3,1,0]]
sage: data_emb += [[-2,5,4,-3,1,2,0,-1],[5,-3,3],[-2,-3,4,3]]
sage: for k in range(6):
....:     embed[k] = data_emb[k]
sage: G.set_embedding(embed)
sage: TIP.from_minimal_schnyder_wood(G)
The Tamari interval of size 6 induced by relations [(1, 4), (2, 4), (3, 4), (5, 6), (6, 4), (5, 4), (3, 1), (2, 1)]


An example from page 18 of [BeBo2009]:

sage: c0 = [(0,-1),(1,0),(2,-1),(3,2),(4,2),(5,-1)]
sage: c1 = [(5,-2),(2,-2),(4,-2),(3,4),(1,2),(0,2)]
sage: c2 = [(0,-3),(1,-3),(3,-3),(4,-3),(2,-3),(5,2)]
sage: ed = [(u,v,0) for u,v in c0]
sage: ed += [(u,v,1) for u,v in c1]
sage: ed += [(u,v,2) for u,v in c2]
sage: G = DiGraph(ed, format='list_of_edges')
sage: embed = {-1:[5,2,0],-2:[4,2,5],-3:[0,1,2,3,4]}
sage: data_emb = [[2,1,-3,-1],[2,-3,0],[3,-3,1,0,-1,5,-2,4]]
sage: data_emb += [[4,-3,2],[-2,-3,3,2],[-2,2,-1]]
sage: for k in range(6):
....:     embed[k] = data_emb[k]
sage: G.set_embedding(embed)
sage: TIP.from_minimal_schnyder_wood(G)
The Tamari interval of size 6 induced by relations [(1, 3), (2, 3), (4, 5), (5, 3), (4, 3), (2, 1)]


Another small example:

sage: c0 = [(0,-1),(2,-1),(1,0)]
sage: c1 = [(2,-2),(1,-2),(0,2)]
sage: c2 = [(0,-3),(1,-3),(2,1)]
sage: ed = [(u,v,0) for u,v in c0]
sage: ed += [(u,v,1) for u,v in c1]
sage: ed += [(u,v,2) for u,v in c2]
sage: G = DiGraph(ed, format='list_of_edges')
sage: embed = {-1:[2,0],-2:[1,2],-3:[0,1]}
sage: data_emb = [[2,1,-3,-1],[-3,0,2,-2],[-2,1,0,-1]]
sage: for k in range(3):
....:     embed[k] = data_emb[k]
sage: G.set_embedding(embed)
sage: TIP.from_minimal_schnyder_wood(G)
The Tamari interval of size 3 induced by relations [(2, 3), (2, 1)]

static initial_forest(element)

Return the initial forest of a binary tree, an interval-poset or a Dyck word.

An initial forest is an interval-poset corresponding to an initial interval of the Tamari lattice, i.e., containing only increasing relations.

It can be constructed from a binary tree by its binary search tree labeling with the rule: $$a$$ precedes $$b$$ in the initial forest iff $$a$$ is in the left subtree of $$b$$ in the binary search tree.

INPUT:

• element – a binary tree, a Dyck word or an interval-poset

EXAMPLES:

sage: ip = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: TamariIntervalPosets.initial_forest(ip)
The Tamari interval of size 4 induced by relations [(1, 2), (2, 3)]


with binary trees:

sage: bt = BinaryTree(); bt
.
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 0 induced by relations []
sage: bt = BinaryTree([]); bt
[., .]
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 1 induced by relations []
sage: bt = BinaryTree([[],None]); bt
[[., .], .]
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 2 induced by relations [(1, 2)]
sage: bt = BinaryTree([None,[]]); bt
[., [., .]]
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 2 induced by relations []
sage: bt = BinaryTree([[],[]]); bt
[[., .], [., .]]
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 3 induced by relations [(1, 2)]
sage: bt = BinaryTree([[None,[[],None]],[]]); bt
[[., [[., .], .]], [., .]]
sage: TamariIntervalPosets.initial_forest(bt)
The Tamari interval of size 5 induced by relations [(1, 4), (2, 3), (3, 4)]


from Dyck words:

sage: dw = DyckWord([1,0])
sage: TamariIntervalPosets.initial_forest(dw)
The Tamari interval of size 1 induced by relations []
sage: dw = DyckWord([1,1,0,1,0,0,1,1,0,0])
sage: TamariIntervalPosets.initial_forest(dw)
The Tamari interval of size 5 induced by relations [(1, 4), (2, 3), (3, 4)]

le(el1, el2)

Poset structure on the set of interval-posets.

The comparison is first by size, then using the cubical coordinates.

cubical_coordinates()

INPUT:

• el1 – an interval-poset
• el2 – an interval-poset

EXAMPLES:

sage: ip1 = TamariIntervalPoset(4,[(1,2),(2,3),(4,3)])
sage: ip2 = TamariIntervalPoset(4,[(1,2),(2,3)])
sage: TamariIntervalPosets().le(ip1,ip2)
False
sage: TamariIntervalPosets().le(ip2,ip1)
True

options(*get_value, **set_value)

Set and display the options for Tamari interval-posets.

If no parameters are set, then the function returns a copy of the options dictionary.

The options to Tamari interval-posets can be accessed as the method TamariIntervalPosets.options() of TamariIntervalPosets and related parent classes.

OPTIONS:

• latex_color_decreasing – (default: red) the default color of decreasing relations when latexed
• latex_color_increasing – (default: blue) the default color of increasing relations when latexed
• latex_hspace – (default: 1) the default difference between horizontal coordinates of vertices when latexed
• latex_line_width_scalar – (default: 0.5) the default value for the line width as amultiple of the tikz scale when latexed
• latex_tikz_scale – (default: 1) the default value for the tikz scale when latexed
• latex_vspace – (default: 1) the default difference between vertical coordinates of vertices when latexed

EXAMPLES:

sage: TIP = TamariIntervalPosets
sage: TIP.options.latex_color_decreasing
red
sage: TIP.options.latex_color_decreasing='green'
sage: TIP.options.latex_color_decreasing
green
sage: TIP.options._reset()
sage: TIP.options.latex_color_decreasing
red


See GlobalOptions for more features of these options.

static recomposition_from_triple(left, right, r)

Recompose an interval-poset from a triple (left, right, r).

For the inverse method, see TamariIntervalPoset.decomposition_to_triple().

INPUT:

• left – an interval-poset
• right – an interval-poset
• r – the parameter of the decomposition, an integer

OUTPUT: an interval-poset

EXAMPLES:

sage: T1 = TamariIntervalPoset(3, [(1, 2), (3, 2)])
sage: T2 = TamariIntervalPoset(4, [(2, 3), (4, 3)])
sage: TamariIntervalPosets.recomposition_from_triple(T1, T2, 2)
The Tamari interval of size 8 induced by relations [(1, 2), (2, 4),
(3, 4), (6, 7), (8, 7), (6, 4), (5, 4), (3, 2)]


REFERENCES:

class sage.combinat.interval_posets.TamariIntervalPosets_all

The enumerated set of all Tamari interval-posets.

Element

alias of TamariIntervalPoset

one()

Return the unit of the monoid.

This is the empty interval poset, of size 0.

EXAMPLES:

sage: TamariIntervalPosets().one()
The Tamari interval of size 0 induced by relations []

class sage.combinat.interval_posets.TamariIntervalPosets_size(size)

The enumerated set of interval-posets of a given size.

cardinality()

The cardinality of self. That is, the number of interval-posets of size $$n$$.

The formula was given in [Cha2008]:

$\frac{2(4n+1)!}{(n+1)!(3n+2)!} = \frac{2}{n(n+1)} \binom{4n+1}{n-1}.$

EXAMPLES:

sage: [TamariIntervalPosets(i).cardinality() for i in range(6)]
[1, 1, 3, 13, 68, 399]

element_class()
random_element()

Return a random Tamari interval of fixed size.

This is obtained by first creating a random rooted planar triangulation, then computing its unique minimal Schnyder wood, then applying a bijection of Bernardi and Bonichon [BeBo2009].

Because the random rooted planar triangulation is chosen uniformly at random, the Tamari interval is also chosen according to the uniform distribution.

EXAMPLES:

sage: T = TamariIntervalPosets(4).random_element()
sage: T.parent()
Interval-posets
sage: u = T.lower_dyck_word(); u   # random
[1, 1, 0, 1, 0, 0, 1, 0]
sage: v = T.lower_dyck_word(); v   # random
[1, 1, 0, 1, 0, 0, 1, 0]
sage: len(u)
8