Identifier
Values
[1] => ([],1) => ([(0,1)],2) => 1
[1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 2
[2,1] => ([(0,1)],2) => ([(0,2),(2,1)],3) => 2
[1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 3
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4
[3,2,1] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) => 6
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9) => 6
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => ([(0,7),(2,12),(3,10),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,4),(7,5),(7,6),(8,13),(9,13),(10,12),(11,3),(11,13),(12,1),(13,2),(13,10)],14) => 7
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => 6
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 7
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 7
[8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 8
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is join irreducible if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
order ideals
Description
The lattice of order ideals of a poset.
An order ideal $\mathcal I$ in a poset $P$ is a downward closed set, i.e., $a \in \mathcal I$ and $b \leq a$ implies $b \in \mathcal I$. This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.