- FindStat

Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000454
(load all 4 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000703

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000329

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St001508

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. Given two lattice paths

$U,L$ from

$(0,0)$ to

$(d,n-d)$ , [1] describes a bijection between lattice paths weakly between

$U$ and

$L$ and subsets of

$\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid

$M[U,L]$ . This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.

Matching statistic: St001232

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 62%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001893

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
St001893: Signed permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%

Values

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

Description

The flag descent of a signed permutation.

$fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 )$ It has the same distribution as the flag excedance statistic.

Matching statistic: St001769
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%

Values

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

Description

The reflection length of a signed permutation. This is the minimal numbers of reflections needed to express a signed permutation.

Matching statistic: St001896
(load all 2 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%

Values

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

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: St001621

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%

Values

[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,4,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,4,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1],[2],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[3],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[3],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1],[2,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1],[2],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[3],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[3],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[3],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1],[2,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[3,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 + 1
[[1,1,1,1],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[2,2,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3 + 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 2 + 1
[[1,1,1,1,2],[2]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2 + 1
[[1,2,2,2,2],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? = 1 + 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2 + 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 3 + 1
[[1],[2],[6]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

Matching statistic: St001624

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 56%●distinct values known / distinct values provided: 33%

Values

[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[2,3],[3]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,1,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,2,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,4,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,2,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[2,4,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[3,3,4]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1],[2],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[3],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[3],[4]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1],[2,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[3,3]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,3],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,1,1,1],[2]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1],[2],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[3],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[3],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[2],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[3],[4],[5]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[[1,1],[2,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,1],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,2],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,2],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[2,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[3,3],[4,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 2 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,4],[2],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1,4],[3],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[2,4],[3],[4]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 + 1
[[1,1,1,1],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,1,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,2,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[2,2,2,2],[3]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(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
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 3 + 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 2 + 1
[[1,1,1,1,2],[2]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? = 2 + 1
[[1,2,2,2,2],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? = 1 + 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? = 2 + 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ? = 3 + 1
[[1],[2],[6]]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .

The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001330The hat guessing number of a graph. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001645The pebbling number of a connected graph. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000736The last entry in the first row of a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition.