Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤ
Values
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3) => 2
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
{{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
{{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 6
{{1},{2,3},{4},{5,6}} => [1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
{{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
{{1},{2,4},{3},{5,6}} => [1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
{{1},{2,4},{3},{5},{6}} => [1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 9
{{1},{2},{3,4},{5,6}} => [1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
{{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
{{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
{{1},{2,3,4,5},{6,7}} => [1,3,4,5,2,7,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
{{1},{2,3,4,5},{6},{7}} => [1,3,4,5,2,6,7] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 8
{{1},{2,3,4},{5},{6,7}} => [1,3,4,2,5,7,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
{{1},{2,3,4},{5},{6},{7}} => [1,3,4,2,5,6,7] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 7
{{1},{2,3,5},{4},{6,7}} => [1,3,5,4,2,7,6] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
{{1},{2,3,5},{4},{6},{7}} => [1,3,5,4,2,6,7] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 10
{{1},{2,3},{4},{5},{6,7}} => [1,3,2,4,5,7,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
{{1},{2,3},{4},{5},{6},{7}} => [1,3,2,4,5,6,7] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
{{1},{2,4,5},{3},{6,7}} => [1,4,3,5,2,7,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
{{1},{2,4,5},{3},{6},{7}} => [1,4,3,5,2,6,7] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 10
{{1},{2,4},{3,5},{6,7}} => [1,4,5,2,3,7,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 8
{{1},{2,4},{3,5},{6},{7}} => [1,4,5,2,3,6,7] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 8
{{1},{2,4},{3},{5},{6,7}} => [1,4,3,2,5,7,6] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 10
{{1},{2,4},{3},{5},{6},{7}} => [1,4,3,2,5,6,7] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 10
{{1},{2,5},{3,4},{6,7}} => [1,5,4,3,2,7,6] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
{{1},{2,5},{3,4},{6},{7}} => [1,5,4,3,2,6,7] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 16
{{1},{2},{3,4,5},{6,7}} => [1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
{{1},{2},{3,4,5},{6},{7}} => [1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 7
{{1},{2},{3,4},{5},{6,7}} => [1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
{{1},{2},{3,4},{5},{6},{7}} => [1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
{{1},{2,5},{3},{4},{6,7}} => [1,5,3,4,2,7,6] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
{{1},{2,5},{3},{4},{6},{7}} => [1,5,3,4,2,6,7] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 12
{{1},{2},{3,5},{4},{6,7}} => [1,2,5,4,3,7,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 10
{{1},{2},{3,5},{4},{6},{7}} => [1,2,5,4,3,6,7] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 10
{{1},{2},{3},{4,5},{6,7}} => [1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
{{1},{2},{3},{4,5},{6},{7}} => [1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
{{1},{2},{3},{4},{5},{6,7}} => [1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
{{1},{2},{3},{4},{5},{6},{7}} => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
{{1},{2},{3},{4},{5},{6},{7},{8}} => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
{{1},{2},{3},{4},{5},{6},{7,8}} => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
{{1},{2},{3,5},{4,6},{7},{8}} => [1,2,5,6,3,4,7,8] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 9
{{1},{2,4},{3,5,6},{7,8}} => [1,4,5,2,6,3,8,7] => [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 9
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!