- FindStat

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Identifier

St001720: Lattices ⟶ ℤ

Values

([],1) => 1

([(0,1)],2) => 2

([(0,2),(2,1)],3) => 3

([(0,1),(0,2),(1,3),(2,3)],4) => 2

([(0,3),(2,1),(3,2)],4) => 4

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3

([(0,4),(2,3),(3,1),(4,2)],5) => 5

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 2

([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 2

([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 2

([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 3

([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3

([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4

([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 2

([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 2

([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 3

([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 2

([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6

([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3

([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2

([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 2

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 2

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 2

([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 3

([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 3

([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 3

([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 4

([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2

([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 3

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 2

([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 5

([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 3

([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 4

([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 2

([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 2

([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 4

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 2

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 2

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 3

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 2

([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 4

([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 3

([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 2

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 2

([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 5

([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 3

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 3

([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 3

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 2

([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 2

([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 3

([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 4

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 4

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2

([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 3

([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 3

([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 3

([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 4

([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 3

([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 3

([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 3

([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 2

([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 5

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 2

([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 4

([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 2

([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 4

([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 3

([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7

([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 3

([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 3

([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 5

([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8) => 2

([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8) => 2

([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8) => 2

([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8) => 2

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8) => 3

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,4),(6,1),(6,2),(6,3),(6,5)],8) => 3

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8) => 3

([(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,4),(7,6)],8) => 3

([(0,6),(1,7),(2,7),(3,7),(4,7),(6,1),(6,2),(6,3),(6,4),(7,5)],8) => 4

([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2),(6,5)],8) => 2

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8) => 3

([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8) => 2

([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(7,6)],8) => 2

([(0,6),(2,7),(3,7),(4,7),(5,1),(6,2),(6,3),(6,4),(7,5)],8) => 5

([(0,5),(1,7),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4),(6,7)],8) => 3

([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8) => 3

([(0,5),(1,7),(2,7),(3,6),(4,3),(4,7),(5,1),(5,2),(5,4),(7,6)],8) => 4

([(0,5),(2,7),(3,6),(4,6),(5,2),(5,3),(5,4),(6,7),(7,1)],8) => 4

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,4),(7,3)],8) => 3

([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(6,1),(6,4),(6,5)],8) => 3

([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8) => 3

([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8) => 2

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Description

The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.

References

[1] Tasoulas, I., Manes, K., Sapounakis, A., Tsikouras, P. Chains with small intervals in the lattice of binary paths MathSciNet:4054761

Code

def statistic(L):
    e = L.bottom()
    t = L.top()
    d = 1
    while e != t:
        U = L.upper_covers(e)
        e = L.join(U)
        d += 1
    return d

Created

May 23, 2021 at 21:44 by Martin Rubey

Updated

May 23, 2021 at 21:44 by Martin Rubey