Identifier
Values
[1,0] => [2,1] => [1,2] => ([(0,1)],2) => 2
[1,0,1,0] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[1,1,0,0] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0] => [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4
[1,0,1,1,0,0] => [3,1,4,2] => [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) => 3
[1,1,0,0,1,0] => [2,4,1,3] => [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) => 3
[1,1,0,1,0,0] => [4,3,1,2] => [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) => 3
[1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
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Description
The number of ordinal summands of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.