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Statistic identifier: St001754

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Collection: Lattices

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Description: The number of tolerances of a finite lattice.

Let $L$ be a lattice.  A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join.  Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$.

The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].

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References: [1]   Chajda, I., Zelinka, B. Tolerance relation on lattices [[MathSciNet:0360380]]
[2]   Bandelt, H.-J. Tolerante Catalanzahlen [[MathSciNet:0725192]]

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Code:


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Statistic values:

([],1)                                                => 1
([(0,1)],2)                                           => 2
([(0,2),(2,1)],3)                                     => 5
([(0,1),(0,2),(1,3),(2,3)],4)                         => 4
([(0,3),(2,1),(3,2)],4)                               => 14
([(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)                   => 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)                   => 13
([(0,4),(2,3),(3,1),(4,2)],5)                         => 42
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)                   => 13
([(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)       => 3
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)       => 3
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)       => 8
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)             => 17
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)             => 48
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)             => 8
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)             => 42
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)       => 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)             => 42
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)       => 8
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)             => 7
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)       => 10
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)                   => 132
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)             => 17
([],0)                                                => 1

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Created: Dec 13, 2021 at 11:51 by Martin Rubey

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Last Updated: Dec 13, 2021 at 11:51 by Martin Rubey