Mobile posets

class sage.combinat.posets.mobile.MobilePoset(hasse_diagram, elements, category, facade, key, ribbon=None, check=True)

Bases: sage.combinat.posets.posets.FinitePoset

A mobile poset.

Mobile posets are an extension of d-complete posets which permit a determinant formula for counting linear extensions. They are formed by having a ribbon poset with d-complete posets ‘hanging’ below it and at most one d-complete poset above it, known as the anchor. See [GGMM2020] for the definition.

EXAMPLES:

sage: P = posets.MobilePoset(posets.RibbonPoset(7, [1,3]),
....:                        {1: [posets.YoungDiagramPoset([3, 2], dual=True)],
....:                         3: [posets.DoubleTailedDiamond(6)]},
....:                        anchor=(4, 2, posets.ChainPoset(6)))
sage: len(P._ribbon)
8
sage: P._anchor
(4, 5)

This example is Example 5.9 in [GGMM2020]:

sage: P1 = posets.MobilePoset(posets.RibbonPoset(8, [2,3,4]),
....:                         {4: [posets.ChainPoset(1)]},
....:                         anchor=(3, 0, posets.ChainPoset(1)))
sage: sorted([P1._element_to_vertex(i) for i in P1._ribbon])
[0, 1, 2, 6, 7, 9]
sage: P1._anchor
(3, 2)

sage: P2 = posets.MobilePoset(posets.RibbonPoset(15, [1,3,5,7,9,11,13]),
....:                         {}, anchor=(8, 0, posets.ChainPoset(1)))
sage: sorted(P2._ribbon)
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
sage: P2._anchor
(8, (8, 0))
sage: P2.linear_extensions().cardinality()
21399440939

sage: EP = posets.MobilePoset(posets.ChainPoset(0), {})
Traceback (most recent call last):
...
ValueError: the empty poset is not a mobile poset

anchor()

Return the anchor of the mobile poset.

EXAMPLES:

sage: from sage.combinat.posets.mobile import MobilePoset
sage: M2 = MobilePoset(Poset([[0,1,2,3,4,5,6,7,8],
....:          [(1,0),(3,0),(2,1),(2,3),(4,3),(5,4),(7,4),(7,8)]]))
sage: M2.anchor()
(4, 3)
sage: M3 = MobilePoset(Posets.RibbonPoset(5, [1,2]))
sage: M3.anchor() is None
True

ribbon()

Return the ribbon of the mobile poset.

EXAMPLES:

sage: from sage.combinat.posets.mobile import MobilePoset
sage: M3 = MobilePoset(Posets.RibbonPoset(5, [1,2]))
sage: sorted(M3.ribbon())
[1, 2, 3, 4]