Identifier
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [2,1] => ([],2) => 0
{{1},{2}} => [1,2] => [1,2] => ([(0,1)],2) => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => ([],3) => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
{{1,2,4},{3}} => [2,4,3,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
{{1,3,4},{2}} => [3,2,4,1] => [4,3,2,1] => ([],4) => 0
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 0
{{1,4},{2,3}} => [4,3,2,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5) => 0
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
{{1,3,4,5},{2}} => [3,2,4,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5) => 0
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 2
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [3,4,5,2,1] => ([(2,3),(3,4)],5) => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 0
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 2
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 2
{{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 1
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6) => 1
{{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 1
{{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6) => 1
{{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
{{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6) => 0
{{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 1
{{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 3
{{1,3},{2,4,5},{6}} => [3,4,1,5,2,6] => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,3},{2,5},{4},{6}} => [3,5,1,4,2,6] => [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
{{1,3},{2,6},{4,5}} => [3,6,1,5,4,2] => [4,5,3,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => 0
{{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 1
{{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,4,5},{2,3},{6}} => [4,3,2,5,1,6] => [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,4,6},{2,3},{5}} => [4,3,2,6,5,1] => [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 1
{{1,6},{2,3,4},{5}} => [6,3,4,2,5,1] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => 0
{{1,5},{2,3},{4},{6}} => [5,3,2,4,1,6] => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
{{1,6},{2,3},{4,5}} => [6,3,2,5,4,1] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6) => 1
{{1,6},{2,3},{4},{5}} => [6,3,2,4,5,1] => [5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6) => 0
{{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,4,5},{2},{3},{6}} => [4,2,3,5,1,6] => [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
{{1,4},{2},{3,5},{6}} => [4,2,5,1,3,6] => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
{{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => [5,4,2,3,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,4},{2},{3,6},{5}} => [4,2,6,1,5,3] => [5,6,4,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 0
{{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => [3,4,2,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 2
{{1,5,6},{2,4},{3}} => [5,4,3,2,6,1] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => 0
{{1,5},{2,4},{3},{6}} => [5,4,3,2,1,6] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 0
{{1,6},{2,4},{3},{5}} => [6,4,3,2,5,1] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,5},{2},{3,4},{6}} => [5,2,4,3,1,6] => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => 0
{{1,6},{2},{3,4},{5}} => [6,2,4,3,5,1] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6) => 0
>>> Load all 107 entries. <<<
{{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
{{1,5},{2},{3},{4,6}} => [5,2,3,6,1,4] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
{{1,6},{2},{3,5},{4}} => [6,2,5,4,3,1] => [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6) => 0
{{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => [5,6,2,3,4,1] => ([(1,3),(2,4),(4,5)],6) => 0
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.