Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤ
Values
[[1]] => [1] => [1] => ([],1) => 0
[[1,2]] => [1,2] => [2,1] => ([],2) => 1
[[1],[2]] => [2,1] => [1,2] => ([(0,1)],2) => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => ([],3) => 2
[[1,3],[2]] => [2,1,3] => [3,1,2] => ([(1,2)],3) => 1
[[1,2],[3]] => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[[1],[2],[3]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => ([],4) => 3
[[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4) => 2
[[1,2,4],[3]] => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4) => 2
[[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 2
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 1
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4) => 1
[[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 1
[[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 4
[[1,3,4,5],[2]] => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5) => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 3
[[1,2,3,5],[4]] => [4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => 3
[[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5) => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 2
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 2
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 5
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [6,5,4,3,1,2] => ([(4,5)],6) => 4
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 4
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 4
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [6,5,3,1,4,2] => ([(2,5),(3,4),(3,5)],6) => 3
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [6,5,2,1,4,3] => ([(2,4),(2,5),(3,4),(3,5)],6) => 4
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [6,4,3,1,5,2] => ([(1,5),(2,5),(3,4),(3,5)],6) => 3
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [6,4,2,1,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [6,3,2,1,5,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 3
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 3
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 4
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 3
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 3
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6) => 3
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 3
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6) => 3
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6) => 3
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 3
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 3
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 3
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 3
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 3
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 3
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 4
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [6,4,1,5,2,3] => ([(1,5),(2,3),(2,5),(3,4)],6) => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6) => 3
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [6,2,1,5,3,4] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6) => 3
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [6,3,1,4,2,5] => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [6,2,1,4,3,5] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [5,4,1,6,2,3] => ([(0,5),(1,5),(2,3),(2,5),(3,4)],6) => 2
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [5,3,1,6,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [5,2,1,6,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6) => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [4,3,1,6,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6) => 3
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [4,2,1,6,3,5] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6) => 3
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6) => 3
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [5,3,1,4,2,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 3
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [4,3,1,5,2,6] => ([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [4,2,1,5,3,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 3
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 2
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6) => 2
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6) => 2
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6) => 2
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 2
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 2
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 2
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 2
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6) => 1
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6) => 2
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Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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