edit this statistic or download as text // json
Identifier
Values
['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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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