- FindStat

Identifier

St001891: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['A',1]=>4
['A',2]=>8
['B',2]=>8
['G',2]=>8
['A',3]=>8
['B',3]=>10
['C',3]=>10
['A',4]=>8
['B',4]=>10
['C',4]=>10
                    

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Description

The size of a smallest Eulerian poset which does not appear as an interval in the Bruhat order of the Weyl group.
A bounded and graded poset is Eulerian if every non-trivial interval has the same number of elements of even and odd rank.
It is known that every interval of a Bruhat order is Eulerian. This statistic yields the minimal cardinality of an Eulerian poset not appearing in the Bruhat order.

Code

def statistic(ct):
    W = WeylGroup(ct)
    P = W.bruhat_poset()
    intervals = set(P.subposet(P.interval(x, y)).canonical_label()
                    for x, y in P.relations_iterator())
    sizes = [i.cardinality() for i in intervals]
    sizes_counts = [sizes.count(i) for i in range(2, max(sizes) + 3, 2)]
    for i, c in enumerate(sizes_counts, 1):
        if c < len(eulerian_posets(2*i)):
            return 2*i

Created

Mar 29, 2023 at 11:51 by Martin Rubey

Updated

Mar 29, 2023 at 11:51 by Martin Rubey