# Enumerated set of lists of integers with constraints, in inverse lexicographic order¶

HISTORY:

This generic tool was originally written by Hivert and Thiery in MuPAD-Combinat in 2000 and ported over to Sage by Mike Hansen in 2007. It was then completely rewritten in 2015 by Gillespie, Schilling, and Thiery, with the help of many, to deal with limitations and lack of robustness w.r.t. input.

class sage.combinat.integer_lists.invlex.IntegerListsBackend_invlex

Cython back-end of an set of lists of integers with specified constraints enumerated in inverse lexicographic order.

check
class sage.combinat.integer_lists.invlex.IntegerListsLex(*args, **kwds)

Lists of nonnegative integers with constraints, in inverse lexicographic order.

An integer list is a list $$l$$ of nonnegative integers, its parts. The slope (at position $$i$$) is the difference l[i+1]-l[i] between two consecutive parts.

This class allows to construct the set $$S$$ of all integer lists $$l$$ satisfying specified bounds on the sum, the length, the slope, and the individual parts, enumerated in inverse lexicographic order, that is from largest to smallest in lexicographic order. Note that, to admit such an enumeration, $$S$$ is almost necessarily finite (see On finiteness and inverse lexicographic enumeration).

The main purpose is to provide a generic iteration engine for all the enumerated sets like Partitions, Compositions, IntegerVectors. It can also be used to generate many other combinatorial objects like Dyck paths, Motzkin paths, etc. Mathematically speaking, this is a special case of set of integral points of a polytope (or union thereof, when the length is not fixed).

INPUT:

• min_sum – a nonnegative integer (default: 0): a lower bound on sum(l).

• max_sum – a nonnegative integer or $$\infty$$ (default: $$\infty$$): an upper bound on sum(l).

• n – a nonnegative integer (optional): if specified, this overrides min_sum and max_sum.

• min_length – a nonnegative integer (default: $$0$$): a lower bound on len(l).

• max_length – a nonnegative integer or $$\infty$$ (default: $$\infty$$): an upper bound on len(l).

• length – an integer (optional); overrides min_length and max_length if specified;

• min_part – a nonnegative integer: a lower bounds on all

parts: min_part <= l[i] for 0 <= i < len(l).

• floor – a list of nonnegative integers or a function: lower bounds on the individual parts $$l[i]$$.

If floor is a list of integers, then floor<=l[i] for 0 <= i < min(len(l), len(floor). Similarly, if floor is a function, then floor(i) <= l[i] for 0 <= i < len(l).

• max_part – a nonnegative integer or $$\infty$$: an upper bound on all parts: l[i] <= max_part for 0 <= i < len(l).

• ceiling – upper bounds on the individual parts l[i]; this takes the same type of input as floor, except that $$\infty$$ is allowed in addition to integers, and the default value is $$\infty$$.

• min_slope – an integer or $$-\infty$$ (default: $$-\infty$$): an lower bound on the slope between consecutive parts: min_slope <= l[i+1]-l[i] for 0 <= i < len(l)-1

• max_slope – an integer or $$+\infty$$ (defaults: $$+\infty$$) an upper bound on the slope between consecutive parts: l[i+1]-l[i] <= max_slope for 0 <= i < len(l)-1

• category – a category (default: FiniteEnumeratedSets)

• check – boolean (default: True): whether to display the warnings raised when functions are given as input to floor or ceiling and the errors raised when there is no proper enumeration.

• name – a string or None (default: None) if set, this will be passed down to Parent.rename() to specify the name of self. It is recommended to use rename method directly because this feature may become deprecated.

• element_constructor – a function (or callable) that creates elements of self from a list. See also Parent.

• element_class – a class for the elements of self (default: $$ClonableArray$$). This merely sets the attribute self.Element. See the examples for details.

Note

When several lists satisfying the constraints differ only by trailing zeroes, only the shortest one is enumerated (and therefore counted). The others are still considered valid. See the examples below.

This feature is questionable. It is recommended not to rely on it, as it may eventually be discontinued.

EXAMPLES:

We create the enumerated set of all lists of nonnegative integers of length $$3$$ and sum $$2$$:

sage: C = IntegerListsLex(2, length=3)
sage: C
Integer lists of sum 2 satisfying certain constraints
sage: C.cardinality()
6
sage: [p for p in C]
[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]

sage: [2, 0, 0] in C
True
sage: [2, 0, 1] in C
False
sage: "a" in C
False
sage: ["a"] in C
False
sage: C.first()
[2, 0, 0]


One can specify lower and upper bounds on each part:

sage: list(IntegerListsLex(5, length=3, floor=[1,2,0], ceiling=[3,2,3]))
[[3, 2, 0], [2, 2, 1], [1, 2, 2]]


When the length is fixed as above, one can also use IntegerVectors:

sage: IntegerVectors(2,3).list()
[[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]


Using the slope condition, one can generate integer partitions (but see Partitions):

sage: list(IntegerListsLex(4, max_slope=0))
[, [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]


The following is the list of all partitions of $$7$$ with parts at least $$2$$:

sage: list(IntegerListsLex(7, max_slope=0, min_part=2))
[, [5, 2], [4, 3], [3, 2, 2]]


floor and ceiling conditions

Next we list all partitions of $$5$$ of length at most $$3$$ which are bounded below by [2,1,1]:

sage: list(IntegerListsLex(5, max_slope=0, max_length=3, floor=[2,1,1]))
[, [4, 1], [3, 2], [3, 1, 1], [2, 2, 1]]


Note that  is considered valid, because the floor constraints only apply to existing positions in the list. To obtain instead the partitions containing [2,1,1], one needs to use min_length or length:

sage: list(IntegerListsLex(5, max_slope=0, length=3, floor=[2,1,1]))
[[3, 1, 1], [2, 2, 1]]


Here is the list of all partitions of $$5$$ which are contained in [3,2,2]:

sage: list(IntegerListsLex(5, max_slope=0, max_length=3, ceiling=[3,2,2]))
[[3, 2], [3, 1, 1], [2, 2, 1]]


This is the list of all compositions of $$4$$ (but see Compositions):

sage: list(IntegerListsLex(4, min_part=1))
[, [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]


This is the list of all integer vectors of sum $$4$$ and length $$3$$:

sage: list(IntegerListsLex(4, length=3))
[[4, 0, 0], [3, 1, 0], [3, 0, 1], [2, 2, 0], [2, 1, 1],
[2, 0, 2], [1, 3, 0], [1, 2, 1], [1, 1, 2], [1, 0, 3],
[0, 4, 0], [0, 3, 1], [0, 2, 2], [0, 1, 3], [0, 0, 4]]


For whatever it is worth, the floor and min_part constraints can be combined:

sage: L = IntegerListsLex(5, floor=[2,0,2], min_part=1)
sage: L.list()
[, [4, 1], [3, 2], [2, 3], [2, 1, 2]]


This is achieved by updating the floor upon constructing L:

sage: [L.floor(i) for i in range(5)]
[2, 1, 2, 1, 1]


Similarly, the ceiling and max_part constraints can be combined:

sage: L = IntegerListsLex(4, ceiling=[2,3,1], max_part=2, length=3)
sage: L.list()
[[2, 2, 0], [2, 1, 1], [1, 2, 1]]
sage: [L.ceiling(i) for i in range(5)]
[2, 2, 1, 2, 2]


This can be used to generate Motzkin words (see Wikipedia article Motzkin_number):

sage: def motzkin_words(n):
....:     return IntegerListsLex(length=n+1, min_slope=-1, max_slope=1,
....:                ceiling=+[+oo for i in range(n-1)]+)
sage: motzkin_words(4).list()
[[0, 1, 2, 1, 0],
[0, 1, 1, 1, 0],
[0, 1, 1, 0, 0],
[0, 1, 0, 1, 0],
[0, 1, 0, 0, 0],
[0, 0, 1, 1, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 0],
[0, 0, 0, 0, 0]]
sage: [motzkin_words(n).cardinality() for n in range(8)]
[1, 1, 2, 4, 9, 21, 51, 127]
sage: oeis(_)                  # optional -- internet
0: A001006: Motzkin numbers: number of ways of drawing any number
of nonintersecting chords joining n (labeled) points on a circle.
1: ...
2: ...


or Dyck words (see also DyckWords), through the bijection with paths from $$(0,0)$$ to $$(n,n)$$ with left and up steps that remain below the diagonal:

sage: def dyck_words(n):
....:     return IntegerListsLex(length=n, ceiling=list(range(n+1)), min_slope=0)
sage: [dyck_words(n).cardinality() for n in range(8)]
[1, 1, 2, 5, 14, 42, 132, 429]
sage: dyck_words(3).list()
[[0, 1, 2], [0, 1, 1], [0, 0, 2], [0, 0, 1], [0, 0, 0]]


On finiteness and inverse lexicographic enumeration

The set of all lists of integers cannot be enumerated in inverse lexicographic order, since there is no largest list (take $$[n]$$ for $$n$$ as large as desired):

sage: IntegerListsLex().first()
Traceback (most recent call last):
...
ValueError: could not prove that the specified constraints yield a finite set


Here is a variant which could be enumerated in lexicographic order but not in inverse lexicographic order:

sage: L = IntegerListsLex(length=2, ceiling=[Infinity, 0], floor=[0,1])
sage: for l in L: print(l)
Traceback (most recent call last):
...
ValueError: infinite upper bound for values of m


Even when the sum is specified, it is not necessarily possible to enumerate all elements in inverse lexicographic order. In the following example, the list [1, 1, 1] will never appear in the enumeration:

sage: IntegerListsLex(3).first()
Traceback (most recent call last):
...
ValueError: could not prove that the specified constraints yield a finite set


If one wants to proceed anyway, one can sign a waiver by setting check=False (again, be warned that some valid lists may never appear):

sage: L = IntegerListsLex(3, check=False)
sage: it = iter(L)
sage: [next(it) for i in range(6)]
[, [2, 1], [2, 0, 1], [2, 0, 0, 1], [2, 0, 0, 0, 1], [2, 0, 0, 0, 0, 1]]


In fact, being inverse lexicographically enumerable is almost equivalent to being finite. The only infinity that can occur would be from a tail of numbers $$0,1$$ as in the previous example, where the $$1$$ moves further and further to the right. If there is any list that is inverse lexicographically smaller than such a configuration, the iterator would not reach it and hence would not be considered iterable. Given that the infinite cases are very specific, at this point only the finite cases are supported (without signing the waiver).

The finiteness detection is not complete yet, so some finite cases may not be supported either, at least not without disabling the checks. Practical examples of such are welcome.

On trailing zeroes, and their caveats

As mentioned above, when several lists satisfying the constraints differ only by trailing zeroes, only the shortest one is listed:

sage: L = IntegerListsLex(max_length=4, max_part=1)
sage: L.list()
[[1, 1, 1, 1],
[1, 1, 1],
[1, 1, 0, 1],
[1, 1],
[1, 0, 1, 1],
[1, 0, 1],
[1, 0, 0, 1],
,
[0, 1, 1, 1],
[0, 1, 1],
[0, 1, 0, 1],
[0, 1],
[0, 0, 1, 1],
[0, 0, 1],
[0, 0, 0, 1],
[]]


and counted:

sage: L.cardinality()
16


Still, the others are considered as elements of $$L$$:

sage: L = IntegerListsLex(4,min_length=3,max_length=4)
sage: L.list()
[..., [2, 2, 0], ...]

sage: [2, 2, 0] in L       # in L.list()
True
sage: [2, 2, 0, 0] in L    # not in L.list() !
True
sage: [2, 2, 0, 0, 0] in L
False


Specifying functions as input for the floor or ceiling

We construct all lists of sum $$4$$ and length $$4$$ such that l[i] <= i:

sage: list(IntegerListsLex(4, length=4, ceiling=lambda i: i, check=False))
[[0, 1, 2, 1], [0, 1, 1, 2], [0, 1, 0, 3], [0, 0, 2, 2], [0, 0, 1, 3]]


Warning

When passing a function as floor or ceiling, it may become undecidable to detect improper inverse lexicographic enumeration. For example, the following example has a finite enumeration:

sage: L = IntegerListsLex(3, floor=lambda i: 1 if i>=2 else 0, check=False)
sage: L.list()
[,
[2, 1],
[2, 0, 1],
[1, 2],
[1, 1, 1],
[1, 0, 2],
[1, 0, 1, 1],
[0, 3],
[0, 2, 1],
[0, 1, 2],
[0, 1, 1, 1],
[0, 0, 3],
[0, 0, 2, 1],
[0, 0, 1, 2],
[0, 0, 1, 1, 1]]


but one cannot decide whether the following has an improper inverse lexicographic enumeration without computing the floor all the way to Infinity:

sage: L = IntegerListsLex(3, floor=lambda i: 0, check=False)
sage: it = iter(L)
sage: [next(it) for i in range(6)]
[, [2, 1], [2, 0, 1], [2, 0, 0, 1], [2, 0, 0, 0, 1], [2, 0, 0, 0, 0, 1]]


Hence a warning is raised when a function is specified as input, unless the waiver is signed by setting check=False:

sage: L = IntegerListsLex(3, floor=lambda i: 1 if i>=2 else 0)
doctest:...
A function has been given as input of the floor=[...] or ceiling=[...]
arguments of IntegerListsLex. Please see the documentation for the caveats.
If you know what you are doing, you can set check=False to skip this warning.


Similarly, the algorithm may need to search forever for a solution when the ceiling is ultimately zero:

sage: L = IntegerListsLex(2,ceiling=lambda i:0, check=False)
sage: L.first()           # not tested: will hang forever
sage: L = IntegerListsLex(2,ceiling=lambda i:0 if i<20 else 1, check=False)
sage: it = iter(L)
sage: next(it)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]
sage: next(it)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1]
sage: next(it)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1]


Tip: using disjoint union enumerated sets for additional flexibility

Sometimes, specifying a range for the sum or the length may be too restrictive. One would want instead to specify a list, or iterable $$L$$, of acceptable values. This is easy to achieve using a disjoint union of enumerated sets. Here we want to accept the values $$n=0,2,3$$:

sage: C = DisjointUnionEnumeratedSets(Family([0,2,3],
....:         lambda n: IntegerListsLex(n, length=2)))
sage: C
Disjoint union of Finite family
{0: Integer lists of sum 0 satisfying certain constraints,
2: Integer lists of sum 2 satisfying certain constraints,
3: Integer lists of sum 3 satisfying certain constraints}
sage: C.list()
[[0, 0],
[2, 0], [1, 1], [0, 2],
[3, 0], [2, 1], [1, 2], [0, 3]]


The price to pay is that the enumeration order is now graded lexicographic instead of lexicographic: first choose the value according to the order specified by $$L$$, and use lexicographic order within each value. Here is we reverse $$L$$:

sage: DisjointUnionEnumeratedSets(Family([3,2,0],
....:     lambda n: IntegerListsLex(n, length=2))).list()
[[3, 0], [2, 1], [1, 2], [0, 3],
[2, 0], [1, 1], [0, 2],
[0, 0]]


Note that if a given value appears several times, the corresponding elements will be enumerated several times, which may, or not, be what one wants:

sage: DisjointUnionEnumeratedSets(Family([2,2],
....:     lambda n: IntegerListsLex(n, length=2))).list()
[[2, 0], [1, 1], [0, 2], [2, 0], [1, 1], [0, 2]]


Here is a variant where we specify acceptable values for the length:

sage: DisjointUnionEnumeratedSets(Family([0,1,3],
....:     lambda l: IntegerListsLex(2, length=l))).list()
[,
[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]


This technique can also be useful to obtain a proper enumeration on infinite sets by using a graded lexicographic enumeration:

sage: C = DisjointUnionEnumeratedSets(Family(NN,
....:         lambda n: IntegerListsLex(n, length=2)))
sage: C
Disjoint union of Lazy family (<lambda>(i))_{i in Non negative integer semiring}
sage: it = iter(C)
sage: [next(it) for i in range(10)]
[[0, 0],
[1, 0], [0, 1],
[2, 0], [1, 1], [0, 2],
[3, 0], [2, 1], [1, 2], [0, 3]]


Specifying how to construct elements

This is the list of all monomials of degree $$4$$ which divide the monomial $$x^3y^1z^2$$ (a monomial being identified with its exponent vector):

sage: R.<x,y,z> = QQ[]
sage: m = [3,1,2]
sage: def term(exponents):
....:     return x^exponents * y^exponents * z^exponents
sage: list( IntegerListsLex(4, length=len(m), ceiling=m, element_constructor=term) )
[x^3*y, x^3*z, x^2*y*z, x^2*z^2, x*y*z^2]


Note the use of the element_constructor option to specify how to construct elements from a plain list.

A variant is to specify a class for the elements. With the default element constructor, this class should take as input the parent self and a list.

Warning

The protocol for specifying the element class and constructor is subject to changes.

ALGORITHM:

The iteration algorithm uses a depth first search through the prefix tree of the list of integers (see also Lexicographic generation of lists of integers). While doing so, it does some lookahead heuristics to attempt to cut dead branches.

In most practical use cases, most dead branches are cut. Then, roughly speaking, the time needed to iterate through all the elements of $$S$$ is proportional to the number of elements, where the proportion factor is controlled by the length $$l$$ of the longest element of $$S$$. In addition, the memory usage is also controlled by $$l$$, which is to say negligible in practice.

Still, there remains much room for efficiency improvements; see trac ticket #18055, trac ticket #18056.

Note

The generation algorithm could in principle be extended to deal with non-constant slope constraints and with negative parts.

TESTS from comments on trac ticket #17979

Comment 191:

sage: list(IntegerListsLex(1, min_length=2, min_slope=0, max_slope=0))
[]


Comment 240:

sage: L = IntegerListsLex(min_length=2, max_part=0)
sage: L.list()
[[0, 0]]


Tests on the element constructor feature and mutability

Internally, the iterator works on a single list that is mutated along the way. Therefore, you need to make sure that the element_constructor actually copies its input. This example shows what can go wrong:

sage: P = IntegerListsLex(n=3, max_slope=0, min_part=1, element_constructor=lambda x: x)
sage: list(P)
[[], [], []]


However, specifying list() as constructor solves this problem:

sage: P = IntegerListsLex(n=3, max_slope=0, min_part=1, element_constructor=list)
sage: list(P)
[, [2, 1], [1, 1, 1]]


Same, step by step:

sage: it = iter(P)
sage: a = next(it); a

sage: b = next(it); b
[2, 1]
sage: a

sage: a is b
False


sage: IntegerListsLex(7, min_length=2, max_length=6, floor=[0,0,2,0,0,1], ceiling=[3,2,3,2,1,2]).cardinality()
83
sage: IntegerListsLex(7, min_length=2, max_length=6, floor=[0,0,2,0,1,1], ceiling=[3,2,3,2,1,2]).cardinality()
53
sage: IntegerListsLex(5, min_length=2, max_length=6, floor=[0,0,2,0,0,0], ceiling=[2,2,2,2,2,2]).cardinality()
30
sage: IntegerListsLex(5, min_length=2, max_length=6, floor=[0,0,1,1,0,0], ceiling=[2,2,2,2,2,2]).cardinality()
43

sage: IntegerListsLex(0, min_length=0, max_length=7, floor=[1,1,0,0,1,0], ceiling=[4,3,2,3,2,2,1]).first()
[]

sage: IntegerListsLex(0, min_length=1, max_length=7, floor=[0,1,0,0,1,0], ceiling=[4,3,2,3,2,2,1]).first()

sage: IntegerListsLex(0, min_length=1, max_length=7, floor=[1,1,0,0,1,0], ceiling=[4,3,2,3,2,2,1]).cardinality()
0

sage: IntegerListsLex(2, min_length=0, max_length=7, floor=[1,1,0,0,0,0], ceiling=[4,3,2,3,2,2,1]).first()  # Was [1,1], due to slightly different specs

sage: IntegerListsLex(1, min_length=1, max_length=7, floor=[1,1,0,0,0,0], ceiling=[4,3,2,3,2,2,1]).first()

sage: IntegerListsLex(1, min_length=2, max_length=7, floor=[1,1,0,0,0,0], ceiling=[4,3,2,3,2,2,1]).cardinality()
0
sage: IntegerListsLex(2, min_length=5, max_length=7, floor=[1,1,0,0,0,0], ceiling=[4,3,2,3,2,2,1]).first()
[1, 1, 0, 0, 0]
sage: IntegerListsLex(2, min_length=5, max_length=7, floor=[1,1,0,0,0,1], ceiling=[4,3,2,3,2,2,1]).first()
[1, 1, 0, 0, 0]
sage: IntegerListsLex(2, min_length=5, max_length=7, floor=[1,1,0,0,1,0], ceiling=[4,3,2,3,2,2,1]).cardinality()
0

sage: IntegerListsLex(4, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).cardinality()
0
sage: IntegerListsLex(5, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).first()
[2, 1, 2]
sage: IntegerListsLex(6, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).first()
[3, 1, 2]
sage: IntegerListsLex(12, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).first()
[3, 1, 2, 3, 2, 1]
sage: IntegerListsLex(13, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).first()
[3, 1, 2, 3, 2, 2]
sage: IntegerListsLex(14, min_length=3, max_length=6, floor=[2, 1, 2, 1, 1, 1], ceiling=[3, 1, 2, 3, 2, 2]).cardinality()
0


This used to hang (see comment 389 and fix in Envelope.__init__()):

sage: IntegerListsLex(7, max_part=0, ceiling=lambda i:i, check=False).list()
[]

backend_class
class sage.combinat.integer_lists.invlex.IntegerListsLexIter(backend)

Bases: future.types.newobject.newobject

Iterator class for IntegerListsLex.

Let T be the prefix tree of all lists of nonnegative integers that satisfy all constraints except possibly for min_length and min_sum; let the children of a list be sorted decreasingly according to their last part.

The iterator is based on a depth-first search exploration of a subtree of this tree, trying to cut branches that do not contain a valid list. Each call of next iterates through the nodes of this tree until it finds a valid list to return.

Here are the attributes describing the current state of the iterator, and their invariants:

• backend – the IntegerListsBackend object this is iterating on;
• _current_list – the list corresponding to the current node of the tree;
• _j – the index of the last element of _current_list: self._j == len(self._current_list) - 1;
• _current_sum – the sum of the parts of _current_list;
• _search_ranges – a list of same length as _current_list: the range for each part.

Furthermore, we assume that there is no obvious contradiction in the contraints:

• self.backend.min_length <= self.backend.max_length;
• self.backend.min_slope <= self.backend.max_slope unless self.backend.min_length <= 1.

Along this iteration, next switches between the following states:

• LOOKAHEAD: determine whether the current list could be a prefix of a valid list;
• PUSH: go deeper into the prefix tree by appending the largest possible part to the current list;
• ME: check whether the current list is valid and if yes return it
• DECREASE: decrease the last part;
• POP: pop the last part of the current list;
• STOP: the iteration is finished.

The attribute _next_state contains the next state next should enter in.