Identifier
Values
[[1],[2],[3]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[2],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[3],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[3],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[2],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[3],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[3],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[3],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[3],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[4],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[2],[3],[5]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[4],[5]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[3],[4],[5]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[2],[3],[4],[5]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[3],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[3],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[2],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[3],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[4],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[4],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[5],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[[1],[2],[3],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[4],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[3],[4],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[3],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[4],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[2],[3],[4],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[2],[3],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[2],[4],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[3],[4],[5],[6]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).