- FindStat

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000454

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [1,2] => [1] => ([],1)
 => 0
[[1],[2]]
 => [2,1] => [1] => ([],1)
 => 0
[[1,2,3]]
 => [1,2,3] => [1,2] => ([],2)
 => 0
[[1,3],[2]]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => ([],2)
 => 0
[[1],[2],[3]]
 => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3] => ([],3)
 => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,1,3] => ([(1,2)],3)
 => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
 => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 2
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([],6)
 => 0
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 1
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St001557
(load all 4 compositions to match this statistic)

Mapping

Mp00155: Standard tableaux —promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 57%

Values

[[1,2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1],[2]]
 => [[1],[2]]
 => [2,1] => [1,2] => 0
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [2,1,3] => [1,2,3] => 0
[[1],[2],[3]]
 => [[1],[2],[3]]
 => [3,2,1] => [1,3,2] => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [1,3,2,4] => 1
[[1,2,3],[4]]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,3,2,4] => 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,2,3] => 2
[[1,3],[2],[4]]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [1,3,2,4] => 1
[[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,4,2,3] => 2
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,2,4,5] => 1
[[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,2,3,4,5] => 0
[[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,4,2,5,3] => 2
[[1,3,4],[2,5]]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,3,2,4,5] => 1
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,2,3,5] => 2
[[1,3,4],[2],[5]]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,3,2,4,5] => 1
[[1,4],[2,5],[3]]
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,4,2,3,5] => 2
[[1,2],[3,4],[5]]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,4,2,3,5] => 2
[[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,2,4,3] => 3
[[1,4],[2],[3],[5]]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,4,2,3,5] => 2
[[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,5,2,4,3] => 3
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6],[2]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,2,3,4,6],[5]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ? = 2
[[1,2,3,4,5],[6]]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,2,5,6],[3,4]]
 => [[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [1,4,2,5,3,6] => ? = 2
[[1,3,4,5],[2,6]]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,2,3,4],[5,6]]
 => [[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [1,2,6,5,4,3] => ? = 2
[[1,4,5,6],[2],[3]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [1,4,2,3,5,6] => ? = 2
[[1,2,3,6],[4],[5]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [1,6,4,2,5,3] => ? = 3
[[1,3,4,5],[2],[6]]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,2,3,4],[5],[6]]
 => [[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [1,6,5,4,3,2] => ? = 2
[[1,2,5],[3,4,6]]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 2
[[1,4,5],[2,6],[3]]
 => [[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => [1,4,2,3,5,6] => ? = 2
[[1,2,3],[4,6],[5]]
 => [[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => [1,6,4,2,3,5] => ? = 3
[[1,2,5],[3,4],[6]]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,4,2,3,5,6] => ? = 2
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [1,5,2,4,3,6] => ? = 3
[[1,4,5],[2],[3],[6]]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [1,4,2,3,5,6] => ? = 2
[[1,2,3],[4],[5],[6]]
 => [[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [1,6,4,2,5,3] => ? = 3
[[1,5],[2,6],[3],[4]]
 => [[1,2],[3,6],[4],[5]]
 => [5,4,3,6,1,2] => [1,5,2,4,6,3] => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [1,6,2,5,3,4] => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [1,5,2,4,3,6] => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,6,2,5,3,4] => ? = 4
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,3,4,5,6,7],[2]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 2
[[1,2,3,4,5,6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [1,4,2,5,3,6,7] => ? = 2
[[1,3,4,5,6],[2,7]]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [1,3,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5,7]]
 => [[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [1,2,6,5,4,3,7] => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [1,4,2,3,5,6,7] => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [1,6,4,2,5,3,7] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 2
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [1,5,2,6,3,7,4] => ? = 3
[[1,2,5,6],[3,4,7]]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [1,4,2,5,3,6,7] => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [[1,2,6,7],[3,5],[4]]
 => [4,3,5,1,2,6,7] => [1,4,2,3,5,6,7] => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [[1,3,4,7],[2,5],[6]]
 => [6,2,5,1,3,4,7] => [1,6,4,2,3,5,7] => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [1,4,2,3,5,6,7] => ? = 2
[[1,5,6,7],[2],[3],[4]]
 => [[1,2,6,7],[3],[4],[5]]
 => [5,4,3,1,2,6,7] => [1,5,2,4,3,6,7] => ? = 3
[[1,4,5,6],[2],[3],[7]]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [1,4,2,3,5,6,7] => ? = 2
[[1,2,3,6],[4],[5],[7]]
 => [[1,3,4,7],[2],[5],[6]]
 => [6,5,2,1,3,4,7] => [1,6,4,2,5,3,7] => ? = 3
[[1,2,3],[4,5,6],[7]]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [1,5,2,3,6,4,7] => ? = 3
[[1,2,7],[3,4],[5,6]]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [1,6,2,7,3,4,5] => ? = 4
[[1,5,6],[2,7],[3],[4]]
 => [[1,2,7],[3,6],[4],[5]]
 => [5,4,3,6,1,2,7] => [1,5,2,4,6,3,7] => ? = 3
[[1,6,7],[2],[3],[4],[5]]
 => [[1,2,7],[3],[4],[5],[6]]
 => [6,5,4,3,1,2,7] => [1,6,2,5,3,4,7] => ? = 4
[[1,5,6],[2],[3],[4],[7]]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [1,5,2,4,3,6,7] => ? = 3
[[1,2],[3,4],[5,6],[7]]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [1,6,2,4,3,7,5] => ? = 4
[[1,6],[2,7],[3],[4],[5]]
 => [[1,2],[3,7],[4],[5],[6]]
 => [6,5,4,3,7,1,2] => [1,6,2,5,7,3,4] => ? = 4
[[1,7],[2],[3],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [1,7,2,6,3,5,4] => ? = 5
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? = 4

Description

The number of inversions of the second entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St001960
(load all 3 compositions to match this statistic)

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 57%

Values

[[1,2]]
 => [[1],[2]]
 => [2,1] => [1,2] => 0
[[1],[2]]
 => [[1,2]]
 => [1,2] => [2,1] => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [2,1,3] => [2,3,1] => 0
[[1],[2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 1
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,4,3] => 1
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,4,1] => 0
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 1
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,4,3,2] => 2
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [2,4,3,1] => 1
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 2
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,5,4] => 1
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [2,3,4,5,1] => 0
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,4,2,5,3] => 2
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 2
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [2,3,5,4,1] => 1
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,5,4,3] => 2
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [4,2,5,3,1] => 2
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,5,4,3,2] => 3
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,5,4,3,1] => 2
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => 3
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [1,2,3,4,6,5] => ? = 1
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [1,3,4,5,6,2] => ? = 2
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [2,3,4,5,6,1] => ? = 0
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => [6,5,2,4,1,3] => [1,2,5,3,6,4] => ? = 2
[[1,3,4,5],[2,6]]
 => [[1,2],[3,6],[4],[5]]
 => [5,4,3,6,1,2] => [2,3,4,1,6,5] => ? = 1
[[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [3,4,5,1,6,2] => ? = 2
[[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [1,2,3,6,5,4] => ? = 2
[[1,2,3,6],[4],[5]]
 => [[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [1,4,5,6,3,2] => ? = 3
[[1,3,4,5],[2],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [2,3,4,6,5,1] => ? = 1
[[1,2,3,4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [3,4,5,6,2,1] => ? = 2
[[1,2,5],[3,4,6]]
 => [[1,3],[2,4],[5,6]]
 => [5,6,2,4,1,3] => [2,1,5,3,6,4] => ? = 2
[[1,4,5],[2,6],[3]]
 => [[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [2,3,1,6,5,4] => ? = 2
[[1,2,3],[4,6],[5]]
 => [[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [4,5,1,6,3,2] => ? = 3
[[1,2,5],[3,4],[6]]
 => [[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => [2,5,3,6,4,1] => ? = 2
[[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [1,2,6,5,4,3] => ? = 3
[[1,4,5],[2],[3],[6]]
 => [[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [2,3,6,5,4,1] => ? = 2
[[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [4,5,6,3,2,1] => ? = 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,6,5,4,3,1] => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 4
[[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 0
[[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => ? = 1
[[1,2,3,4,6,7],[5]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => ? = 2
[[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [2,3,4,5,6,7,1] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => [7,6,5,2,4,1,3] => [1,2,3,6,4,7,5] => ? = 2
[[1,3,4,5,6],[2,7]]
 => [[1,2],[3,7],[4],[5],[6]]
 => [6,5,4,3,7,1,2] => [2,3,4,5,1,7,6] => ? = 1
[[1,2,3,4,6],[5,7]]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [2,4,5,6,1,7,3] => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [1,2,5,6,7,4,3] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [[1,2,7],[3],[4],[5],[6]]
 => [6,5,4,3,1,2,7] => [2,3,4,5,7,6,1] => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [[1,5,7],[2],[3],[4],[6]]
 => [6,4,3,2,1,5,7] => [2,4,5,6,7,3,1] => ? = 2
[[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => [7,3,6,2,5,1,4] => [1,5,2,6,3,7,4] => ? = 3
[[1,2,5,6],[3,4,7]]
 => [[1,3],[2,4],[5,7],[6]]
 => [6,5,7,2,4,1,3] => [2,3,1,6,4,7,5] => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [[1,2,3],[4,7],[5],[6]]
 => [6,5,4,7,1,2,3] => [2,3,4,1,7,6,5] => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [[1,4,5],[2,7],[3],[6]]
 => [6,3,2,7,1,4,5] => [2,5,6,1,7,4,3] => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [[1,3,7],[2,4],[5],[6]]
 => [6,5,2,4,1,3,7] => [2,3,6,4,7,5,1] => ? = 2
[[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 3
[[1,4,5,6],[2],[3],[7]]
 => [[1,2,3,7],[4],[5],[6]]
 => [6,5,4,1,2,3,7] => [2,3,4,7,6,5,1] => ? = 2
[[1,2,3,6],[4],[5],[7]]
 => [[1,4,5,7],[2],[3],[6]]
 => [6,3,2,1,4,5,7] => [2,5,6,7,4,3,1] => ? = 3
[[1,2,3],[4,5,6],[7]]
 => [[1,4,7],[2,5],[3,6]]
 => [3,6,2,5,1,4,7] => [5,2,6,3,7,4,1] => ? = 3
[[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => [7,2,4,6,1,3,5] => [1,6,4,2,7,5,3] => ? = 4
[[1,5,6],[2,7],[3],[4]]
 => [[1,2,3,4],[5,7],[6]]
 => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 3
[[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 4
[[1,5,6],[2],[3],[4],[7]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [2,3,7,6,5,4,1] => ? = 3
[[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => [6,4,2,7,5,3,1] => ? = 4
[[1,6],[2,7],[3],[4],[5]]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? = 4
[[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 5
[[1,6],[2],[3],[4],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? = 4

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Matching statistic: St001526

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 57%

Values

[[1,2]]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 0 + 2
[[1],[2]]
 => [2,1] => [2,1] => [1,1,0,0]
 => 2 = 0 + 2
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 1 + 2
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,3,4],[2]]
 => [2,1,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[1,3],[2,4]]
 => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4 = 2 + 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 2 + 2
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 2 + 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 1 + 2
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 3 + 2
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 3 + 2
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 2 + 2
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 2
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 2 + 2
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 3 + 2
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 2
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 2
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [3,4,6,2,5,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 3 + 2
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 2
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,4,6,5,2,1] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 2
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [4,6,3,5,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 2
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [4,6,5,3,2,1] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 2
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 + 2
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 2
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 2
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 2
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 2
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 2
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 2
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [1,2,5,6,7,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 3 + 2
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 2
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 2
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 2
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [3,4,5,7,1,2,6] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [3,4,5,7,2,6,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [1,2,5,7,4,6,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 2
[[1,2,5,6],[3,4],[7]]
 => [7,3,4,1,2,5,6] => [3,4,5,7,1,6,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 2
[[1,4,5,6],[2],[3],[7]]
 => [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 2
[[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => [4,5,7,1,2,6,3] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 2
[[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,6,7,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[[1,5,6],[2,7],[3],[4]]
 => [4,3,2,7,1,5,6] => [4,5,7,3,6,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 2
[[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 4 + 2
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 2
[[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [5,7,3,6,1,4,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 2
[[1,6],[2,7],[3],[4],[5]]
 => [5,4,3,2,7,1,6] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 2
[[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 5 + 2
[[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 2

Description

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001207
(load all 12 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 57%

Values

[[1,2]]
 => [1,2] => [1] => [1] => ? = 0
[[1],[2]]
 => [2,1] => [1] => [1] => ? = 0
[[1,2,3]]
 => [1,2,3] => [1,2] => [1,2] => 0
[[1,3],[2]]
 => [2,1,3] => [2,1] => [2,1] => 1
[[1,2],[3]]
 => [3,1,2] => [1,2] => [1,2] => 0
[[1],[2],[3]]
 => [3,2,1] => [2,1] => [2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1] => [3,2,1] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2] => [1,4,2,3] => 2
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4] => [3,2,1,4] => 2
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [3,4,1,2] => [1,4,2,3] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 3
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4] => [1,2,3,5,4] => ? = 2
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,1,2,5] => [1,4,2,3,5] => ? = 2
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,1,2,3,4] => [1,2,3,5,4] => ? = 2
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [5,4,1,2,3] => [1,2,5,4,3] => ? = 3
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [5,1,2,3,4] => [1,2,3,5,4] => ? = 2
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [3,4,1,2,5] => [1,4,2,3,5] => ? = 2
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [5,4,1,2,3] => [1,2,5,4,3] => ? = 3
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [3,4,1,2,5] => [1,4,2,3,5] => ? = 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [5,4,1,2,3] => [1,2,5,4,3] => ? = 3
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => [1,2,3,5,4,6] => ? = 2
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,4,1,2,5,6] => [1,4,2,3,5,6] => ? = 2
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [5,1,2,3,4,6] => [1,2,3,5,4,6] => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [5,4,1,2,3,6] => [1,2,5,4,3,6] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [5,1,2,3,4,6] => [1,2,3,5,4,6] => ? = 2
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3] => [1,2,6,3,4,5] => ? = 3
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [3,4,1,2,5,6] => [1,4,2,3,5,6] => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [5,4,1,2,3,6] => [1,2,5,4,3,6] => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [7,3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,4,2,3,5,6] => ? = 2
[[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 3
[[1,4,5,6],[2],[3],[7]]
 => [7,3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 2
[[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => [5,4,1,2,3,6] => [1,2,5,4,3,6] => ? = 3
[[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,2,6,3,4,5] => ? = 3
[[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => ? = 4
[[1,5,6],[2,7],[3],[4]]
 => [4,3,2,7,1,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 3
[[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 4
[[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 3
[[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => ? = 4
[[1,6],[2,7],[3],[4],[5]]
 => [5,4,3,2,7,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 4

Description

The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Matching statistic: St000307

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%

Values

[[1,2]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2 + 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)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 2 + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 2 + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 2 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3 + 1
[[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)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2 + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 2 + 1
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
 => ? = 2 + 1
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 2 + 1
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [4,2,5,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
 => ? = 3 + 1
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 1 + 1
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
 => ? = 2 + 1
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 2 + 1
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
 => ? = 2 + 1
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [5,2,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,19),(1,20),(2,15),(2,16),(2,19),(2,20),(3,9),(3,12),(3,13),(3,19),(4,8),(4,11),(4,13),(4,15),(4,20),(5,7),(5,11),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,16),(6,17),(7,21),(7,25),(7,26),(7,27),(8,21),(8,24),(8,26),(9,24),(9,25),(9,26),(11,18),(11,21),(11,25),(12,18),(12,25),(12,27),(13,18),(13,26),(13,27),(14,21),(14,27),(15,24),(15,25),(15,27),(16,24),(16,26),(17,24),(17,26),(17,27),(18,23),(19,24),(19,27),(20,21),(20,25),(20,26),(20,27),(21,22),(21,23),(22,10),(23,10),(24,22),(25,22),(25,23),(26,22),(26,23),(27,22),(27,23)],28)
 => ? = 3 + 1
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 2 + 1
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 3 + 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
 => ? = 2 + 1
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [5,2,6,3,4,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
 => ? = 3 + 1
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [3,2,6,5,1,4] => ([(0,3),(0,4),(0,5),(0,6),(1,12),(1,19),(2,16),(2,19),(3,8),(3,13),(4,7),(4,9),(4,13),(5,2),(5,9),(5,10),(5,13),(6,1),(6,7),(6,8),(6,10),(7,14),(7,17),(7,19),(8,17),(8,19),(9,14),(9,16),(9,19),(10,12),(10,14),(10,16),(10,17),(12,15),(12,18),(13,16),(13,17),(14,15),(14,18),(15,11),(16,15),(16,18),(17,15),(17,18),(18,11),(19,18)],20)
 => ? = 3 + 1
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [3,4,2,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 4 + 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [3,4,2,6,5,1] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 3 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 4 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 2 + 1
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ([(0,1),(0,2),(0,3),(0,4),(0,7),(1,19),(1,25),(2,17),(2,19),(2,25),(3,16),(3,19),(3,25),(4,15),(4,19),(4,25),(5,8),(5,9),(5,10),(5,24),(6,5),(6,12),(6,13),(6,14),(6,20),(7,6),(7,15),(7,16),(7,17),(7,25),(8,18),(8,22),(9,18),(9,22),(10,18),(10,22),(12,8),(12,23),(12,24),(13,9),(13,23),(13,24),(14,10),(14,23),(14,24),(15,12),(15,20),(15,21),(16,13),(16,20),(16,21),(17,14),(17,20),(17,21),(18,11),(19,21),(20,23),(20,24),(21,23),(22,11),(23,22),(24,18),(24,22),(25,20),(25,21)],26)
 => ? = 2 + 1
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => ?
 => ? = 2 + 1
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
 => ? = 2 + 1
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [4,2,5,3,1,6,7] => ?
 => ? = 3 + 1
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [2,7,6,5,4,3,1] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
 => ? = 1 + 1
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [7,6,4,2,5,3,1] => ?
 => ? = 2 + 1
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ?
 => ? = 3 + 1
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [7,6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,7),(1,19),(1,25),(2,17),(2,19),(2,25),(3,16),(3,19),(3,25),(4,15),(4,19),(4,25),(5,8),(5,9),(5,10),(5,24),(6,5),(6,12),(6,13),(6,14),(6,20),(7,6),(7,15),(7,16),(7,17),(7,25),(8,18),(8,22),(9,18),(9,22),(10,18),(10,22),(12,8),(12,23),(12,24),(13,9),(13,23),(13,24),(14,10),(14,23),(14,24),(15,12),(15,20),(15,21),(16,13),(16,20),(16,21),(17,14),(17,20),(17,21),(18,11),(19,21),(20,23),(20,24),(21,23),(22,11),(23,22),(24,18),(24,22),(25,20),(25,21)],26)
 => ? = 2 + 1
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [2,7,6,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,7),(1,20),(1,22),(2,10),(2,15),(3,15),(3,16),(4,9),(4,15),(4,16),(5,6),(5,13),(5,14),(5,17),(6,1),(6,11),(6,12),(6,18),(7,5),(7,9),(7,10),(7,16),(9,14),(9,17),(9,19),(10,13),(10,19),(11,20),(11,22),(12,20),(12,22),(13,12),(13,21),(14,11),(14,18),(14,21),(15,19),(16,17),(16,19),(17,18),(17,21),(18,20),(18,22),(19,21),(20,8),(21,22),(22,8)],23)
 => ? = 2 + 1
[[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [5,2,4,1,7,6,3] => ?
 => ? = 3 + 1
[[1,2,5,6],[3,4],[7]]
 => [7,3,4,1,2,5,6] => [7,6,5,2,3,4,1] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
 => ? = 2 + 1
[[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
 => ? = 3 + 1
[[1,4,5,6],[2],[3],[7]]
 => [7,3,2,1,4,5,6] => [3,2,7,6,5,4,1] => ([(0,4),(0,5),(0,6),(1,16),(2,7),(2,9),(2,19),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(7,18),(7,20),(9,18),(9,20),(10,15),(10,16),(11,9),(11,17),(12,7),(12,17),(12,19),(13,11),(13,15),(14,12),(14,15),(14,16),(15,17),(15,19),(16,19),(17,18),(17,20),(18,8),(19,20),(20,8)],21)
 => ? = 2 + 1
[[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => ?
 => ? = 3 + 1

Description

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset

$P$ . It sends an order ideal

$I$ to the order ideal generated by the minimal antichain of

$P \setminus I$ .

Matching statistic: St001632

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 29%

Values

[[1,2]]
 => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2 + 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)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 2 + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 2 + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 2 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 3 + 1
[[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)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2 + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 2 + 1
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 1 + 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
 => ? = 2 + 1
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 2 + 1
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [4,2,5,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
 => ? = 3 + 1
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [2,6,5,4,3,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ? = 1 + 1
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
 => ? = 2 + 1
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 2 + 1
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => [2,6,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
 => ? = 2 + 1
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => [5,2,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,19),(1,20),(2,15),(2,16),(2,19),(2,20),(3,9),(3,12),(3,13),(3,19),(4,8),(4,11),(4,13),(4,15),(4,20),(5,7),(5,11),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,16),(6,17),(7,21),(7,25),(7,26),(7,27),(8,21),(8,24),(8,26),(9,24),(9,25),(9,26),(11,18),(11,21),(11,25),(12,18),(12,25),(12,27),(13,18),(13,26),(13,27),(14,21),(14,27),(15,24),(15,25),(15,27),(16,24),(16,26),(17,24),(17,26),(17,27),(18,23),(19,24),(19,27),(20,21),(20,25),(20,26),(20,27),(21,22),(21,23),(22,10),(23,10),(24,22),(25,22),(25,23),(26,22),(26,23),(27,22),(27,23)],28)
 => ? = 3 + 1
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
 => ? = 2 + 1
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [3,2,4,1,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
 => ? = 3 + 1
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => [3,2,6,5,4,1] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
 => ? = 2 + 1
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [5,2,6,3,4,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
 => ? = 3 + 1
[[1,5],[2,6],[3],[4]]
 => [4,3,2,6,1,5] => [3,2,6,5,1,4] => ([(0,3),(0,4),(0,5),(0,6),(1,12),(1,19),(2,16),(2,19),(3,8),(3,13),(4,7),(4,9),(4,13),(5,2),(5,9),(5,10),(5,13),(6,1),(6,7),(6,8),(6,10),(7,14),(7,17),(7,19),(8,17),(8,19),(9,14),(9,16),(9,19),(10,12),(10,14),(10,16),(10,17),(12,15),(12,18),(13,16),(13,17),(14,15),(14,18),(15,11),(16,15),(16,18),(17,15),(17,18),(18,11),(19,18)],20)
 => ? = 3 + 1
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [3,4,2,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 4 + 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [3,4,2,6,5,1] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
 => ? = 3 + 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
 => ? = 4 + 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? = 2 + 1
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ([(0,1),(0,2),(0,3),(0,4),(0,7),(1,19),(1,25),(2,17),(2,19),(2,25),(3,16),(3,19),(3,25),(4,15),(4,19),(4,25),(5,8),(5,9),(5,10),(5,24),(6,5),(6,12),(6,13),(6,14),(6,20),(7,6),(7,15),(7,16),(7,17),(7,25),(8,18),(8,22),(9,18),(9,22),(10,18),(10,22),(12,8),(12,23),(12,24),(13,9),(13,23),(13,24),(14,10),(14,23),(14,24),(15,12),(15,20),(15,21),(16,13),(16,20),(16,21),(17,14),(17,20),(17,21),(18,11),(19,21),(20,23),(20,24),(21,23),(22,11),(23,22),(24,18),(24,22),(25,20),(25,21)],26)
 => ? = 2 + 1
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [7,6,5,4,3,1,2] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => ?
 => ? = 2 + 1
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
 => ? = 2 + 1
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [4,2,5,3,1,6,7] => ?
 => ? = 3 + 1
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [2,7,6,5,4,3,1] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
 => ? = 1 + 1
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [7,6,4,2,5,3,1] => ?
 => ? = 2 + 1
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ?
 => ? = 3 + 1
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [7,6,5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,7),(1,19),(1,25),(2,17),(2,19),(2,25),(3,16),(3,19),(3,25),(4,15),(4,19),(4,25),(5,8),(5,9),(5,10),(5,24),(6,5),(6,12),(6,13),(6,14),(6,20),(7,6),(7,15),(7,16),(7,17),(7,25),(8,18),(8,22),(9,18),(9,22),(10,18),(10,22),(12,8),(12,23),(12,24),(13,9),(13,23),(13,24),(14,10),(14,23),(14,24),(15,12),(15,20),(15,21),(16,13),(16,20),(16,21),(17,14),(17,20),(17,21),(18,11),(19,21),(20,23),(20,24),(21,23),(22,11),(23,22),(24,18),(24,22),(25,20),(25,21)],26)
 => ? = 2 + 1
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [2,7,6,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,7),(1,20),(1,22),(2,10),(2,15),(3,15),(3,16),(4,9),(4,15),(4,16),(5,6),(5,13),(5,14),(5,17),(6,1),(6,11),(6,12),(6,18),(7,5),(7,9),(7,10),(7,16),(9,14),(9,17),(9,19),(10,13),(10,19),(11,20),(11,22),(12,20),(12,22),(13,12),(13,21),(14,11),(14,18),(14,21),(15,19),(16,17),(16,19),(17,18),(17,21),(18,20),(18,22),(19,21),(20,8),(21,22),(22,8)],23)
 => ? = 2 + 1
[[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [5,2,4,1,7,6,3] => ?
 => ? = 3 + 1
[[1,2,5,6],[3,4],[7]]
 => [7,3,4,1,2,5,6] => [7,6,5,2,3,4,1] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
 => ? = 2 + 1
[[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
 => ? = 3 + 1
[[1,4,5,6],[2],[3],[7]]
 => [7,3,2,1,4,5,6] => [3,2,7,6,5,4,1] => ([(0,4),(0,5),(0,6),(1,16),(2,7),(2,9),(2,19),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(7,18),(7,20),(9,18),(9,20),(10,15),(10,16),(11,9),(11,17),(12,7),(12,17),(12,19),(13,11),(13,15),(14,12),(14,15),(14,16),(15,17),(15,19),(16,19),(17,18),(17,20),(18,8),(19,20),(20,8)],21)
 => ? = 2 + 1
[[1,2,3,6],[4],[5],[7]]
 => [7,5,4,1,2,3,6] => [5,2,7,6,3,4,1] => ?
 => ? = 3 + 1

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Matching statistic: St001864

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 43%

Values

[[1,2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1],[2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => [3,1,2] => 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => [3,1,2] => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,2,3,4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,5],[3,4]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[[1,3,4],[2,5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,3,4],[2],[5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,2],[3,4],[5]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6],[2]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4,6],[5]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[[1,2,3,4,5],[6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[1,3,4,5],[2,6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4],[5,6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 2
[[1,4,5,6],[2],[3]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3,6],[4],[5]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 3
[[1,3,4,5],[2],[6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4],[5],[6]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2
[[1,4,5],[2,6],[3]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 3
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1,4,5],[2],[3],[6]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3],[4],[5],[6]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? = 4
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,3,4,5,6,7],[2]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,2,3,4,5,6],[7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,3,4,5,6],[2,7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [6,5,1,2,3,4,7] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 3
[[1,2,5,6],[3,4,7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [6,5,1,2,3,4,7] => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,5,6,7],[2],[3],[4]]
 => [[1,2,6,7],[3],[4],[5]]
 => [5,4,3,1,2,6,7] => [5,4,3,1,2,6,7] => ? = 3
[[1,4,5,6],[2],[3],[7]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2

Description

The number of excedances of a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$ .

Matching statistic: St001896

Mapping

Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 43%

Values

[[1,2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1],[2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => [3,1,2] => 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => [3,1,2] => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [3,1,2,4] => 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,2,3,4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,5],[3,4]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[[1,3,4],[2,5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,3,4],[2],[5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1
[[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1,2],[3,4],[5]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [4,3,1,2,5] => ? = 2
[[1],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 3
[[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6],[2]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4,6],[5]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[[1,2,3,4,5],[6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[1,3,4,5],[2,6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4],[5,6]]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 2
[[1,4,5,6],[2],[3]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3,6],[4],[5]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 3
[[1,3,4,5],[2],[6]]
 => [[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1
[[1,2,3,4],[5],[6]]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 2
[[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2
[[1,4,5],[2,6],[3]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 3
[[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 2
[[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1,4,5],[2],[3],[6]]
 => [[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 2
[[1,2,3],[4],[5],[6]]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 3
[[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? = 4
[[1,2,3,4,5,6,7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,3,4,5,6,7],[2]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6,7],[5]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,2,3,4,5,6],[7]]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[1,2,5,6,7],[3,4]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,3,4,5,6],[2,7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5,7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [6,5,1,2,3,4,7] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 2
[[1,2,3,7],[4,5,6]]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 3
[[1,2,5,6],[3,4,7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [6,5,1,2,3,4,7] => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => ? = 2
[[1,5,6,7],[2],[3],[4]]
 => [[1,2,6,7],[3],[4],[5]]
 => [5,4,3,1,2,6,7] => [5,4,3,1,2,6,7] => ? = 3
[[1,4,5,6],[2],[3],[7]]
 => [[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2

Description

The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.

Matching statistic: St000174

Mapping

Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000174: Semistandard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 43%

Values

[[1,2]]
 => [[2,0],[1]]
 => [[1,2]]
 => [[1,2]]
 => 0
[[1],[2]]
 => [[1,1],[1]]
 => [[1],[2]]
 => [[1,2]]
 => 0
[[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => [[1,2,3]]
 => 0
[[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => [[1,2],[3]]
 => [[1,2,3]]
 => 0
[[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => [[1,2],[3]]
 => 1
[[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4]]
 => [[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => [[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2]]
 => [[1,2,4],[3]]
 => 1
[[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4]]
 => [[1,2,3,4]]
 => 0
[[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => [[1,2,4],[3]]
 => 1
[[1,4],[2],[3]]
 => [[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 2
[[1,3],[2],[4]]
 => [[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 1
[[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 2
[[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => ? = 0
[[1,3,4,5],[2]]
 => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5],[2]]
 => [[1,2,4,5],[3]]
 => ? = 1
[[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4],[5]]
 => [[1,2,3,4,5]]
 => ? = 0
[[1,2,5],[3,4]]
 => [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5],[3,4]]
 => [[1,2,3,4],[5]]
 => ? = 2
[[1,3,4],[2,5]]
 => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2,5]]
 => [[1,2,4,5],[3]]
 => ? = 1
[[1,4,5],[2],[3]]
 => [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5],[2],[3]]
 => [[1,2,5],[3],[4]]
 => ? = 2
[[1,3,4],[2],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2],[5]]
 => [[1,2,4,5],[3]]
 => ? = 1
[[1,4],[2,5],[3]]
 => [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2,5],[3]]
 => [[1,2,5],[3],[4]]
 => ? = 2
[[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4],[5]]
 => [[1,2,3,4],[5]]
 => ? = 2
[[1,5],[2],[3],[4]]
 => [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => ? = 3
[[1,4],[2],[3],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3],[5]]
 => [[1,2,5],[3],[4]]
 => ? = 2
[[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5]]
 => ? = 3
[[1,2,3,4,5,6]]
 => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5,6]]
 => [[1,2,3,4,5,6]]
 => ? = 0
[[1,3,4,5,6],[2]]
 => [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5,6],[2]]
 => [[1,2,4,5,6],[3]]
 => ? = 1
[[1,2,3,4,6],[5]]
 => [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,6],[5]]
 => [[1,2,3,4,5],[6]]
 => ? = 2
[[1,2,3,4,5],[6]]
 => [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5],[6]]
 => [[1,2,3,4,5,6]]
 => ? = 0
[[1,2,5,6],[3,4]]
 => [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5,6],[3,4]]
 => [[1,2,3,4],[5,6]]
 => ? = 2
[[1,3,4,5],[2,6]]
 => [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5],[2,6]]
 => [[1,2,4,5,6],[3]]
 => ? = 1
[[1,2,3,4],[5,6]]
 => [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4],[5,6]]
 => [[1,2,3,4,5,6]]
 => ? = 2
[[1,4,5,6],[2],[3]]
 => [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,5,6],[3],[4]]
 => ? = 2
[[1,2,3,6],[4],[5]]
 => [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,6],[4],[5]]
 => [[1,2,3,4],[5],[6]]
 => ? = 3
[[1,3,4,5],[2],[6]]
 => [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5],[2],[6]]
 => [[1,2,4,5,6],[3]]
 => ? = 1
[[1,2,3,4],[5],[6]]
 => [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4],[5],[6]]
 => [[1,2,3,4,5],[6]]
 => ? = 2
[[1,2,5],[3,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5],[3,4,6]]
 => [[1,2,3,4,6],[5]]
 => ? = 2
[[1,4,5],[2,6],[3]]
 => [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5],[2,6],[3]]
 => [[1,2,5,6],[3],[4]]
 => ? = 2
[[1,2,3],[4,6],[5]]
 => [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4,6],[5]]
 => [[1,2,3,4,6],[5]]
 => ? = 3
[[1,2,5],[3,4],[6]]
 => [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5],[3,4],[6]]
 => [[1,2,3,4],[5,6]]
 => ? = 2
[[1,5,6],[2],[3],[4]]
 => [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => ? = 3
[[1,4,5],[2],[3],[6]]
 => [[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5],[2],[3],[6]]
 => [[1,2,5,6],[3],[4]]
 => ? = 2
[[1,2,3],[4],[5],[6]]
 => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4],[5],[6]]
 => [[1,2,3,4],[5],[6]]
 => ? = 3
[[1,5],[2,6],[3],[4]]
 => [[2,2,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,5],[2,6],[3],[4]]
 => [[1,2,6],[3],[4],[5]]
 => ? = 3
[[1,6],[2],[3],[4],[5]]
 => [[2,1,1,1,1,0],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => ? = 4
[[1,5],[2],[3],[4],[6]]
 => [[2,1,1,1,1,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,5],[2],[3],[4],[6]]
 => [[1,2,6],[3],[4],[5]]
 => ? = 3
[[1],[2],[3],[4],[5],[6]]
 => [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2],[3],[4],[5],[6]]
 => ? = 4
[[1,2,3,4,5,6,7]]
 => [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5,6,7]]
 => ?
 => ? = 0
[[1,3,4,5,6,7],[2]]
 => [[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5,6,7],[2]]
 => ?
 => ? = 1
[[1,2,3,4,6,7],[5]]
 => [[6,1,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ?
 => ?
 => ? = 2
[[1,2,3,4,5,6],[7]]
 => [[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5,6],[7]]
 => ?
 => ? = 0
[[1,2,5,6,7],[3,4]]
 => [[5,2,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5,6,7],[3,4]]
 => ?
 => ? = 2
[[1,3,4,5,6],[2,7]]
 => [[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5,6],[2,7]]
 => ?
 => ? = 1
[[1,2,3,4,6],[5,7]]
 => [[5,2,0,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,6],[5,7]]
 => ?
 => ? = 2
[[1,4,5,6,7],[2],[3]]
 => [[5,1,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5,6,7],[2],[3]]
 => ?
 => ? = 2
[[1,2,3,6,7],[4],[5]]
 => [[5,1,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ?
 => ?
 => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5,6],[2],[7]]
 => ?
 => ? = 1
[[1,2,3,4,6],[5],[7]]
 => [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => ?
 => ?
 => ? = 2
[[1,2,3,7],[4,5,6]]
 => [[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,7],[4,5,6]]
 => ?
 => ? = 3
[[1,2,5,6],[3,4,7]]
 => [[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5,6],[3,4,7]]
 => ?
 => ? = 2
[[1,4,5,6],[2,7],[3]]
 => [[4,2,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => ?
 => ?
 => ? = 2
[[1,2,3,6],[4,7],[5]]
 => [[4,2,1,0,0,0,0],[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => ?
 => ?
 => ? = 3
[[1,2,5,6],[3,4],[7]]
 => [[4,2,1,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5,6],[3,4],[7]]
 => ?
 => ? = 2

Description

The flush statistic of a semistandard tableau. Let

$T$ be a tableaux with

$r$ rows such that each row is longer than the row beneath it by at least one box. Let

$1 \leq i < k \leq r+1$ and suppose

$l$ is the smallest integer greater than

$k$ such that there exists an

$l$ -segment in the

$(i+1)$ -st row of

$T$ . A

$k$ -segment in the

$i$ -th row of

$T$ is called '''flush''' if the leftmost box in the

$k$ -segment and the leftmost box of the

$l$ -segment are in the same column of

$T$ . If, however, no such

$l$ exists, then this

$k$ -segment is said to be flush if the number of boxes in the

$k$ -segment is equal to difference of the number of boxes between the

$i$ -th row and

$(i+1)$ -st row. The flush statistic is given by the number of

$k$ -segments in

$T$ .

The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001946The number of descents in a parking function. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph.