Your data matches 93 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000316
(load all 11 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => 0
[[2,2]]
 => [1,2] => 0
[[1],[2]]
 => [2,1] => 1
[[1,3]]
 => [1,2] => 0
[[2,3]]
 => [1,2] => 0
[[3,3]]
 => [1,2] => 0
[[1],[3]]
 => [2,1] => 1
[[2],[3]]
 => [2,1] => 1
[[1,1,2]]
 => [1,2,3] => 0
[[1,2,2]]
 => [1,2,3] => 0
[[2,2,2]]
 => [1,2,3] => 0
[[1,1],[2]]
 => [3,1,2] => 2
[[1,2],[2]]
 => [2,1,3] => 1
[[1,4]]
 => [1,2] => 0
[[2,4]]
 => [1,2] => 0
[[3,4]]
 => [1,2] => 0
[[4,4]]
 => [1,2] => 0
[[1],[4]]
 => [2,1] => 1
[[2],[4]]
 => [2,1] => 1
[[3],[4]]
 => [2,1] => 1
[[1,1,3]]
 => [1,2,3] => 0
[[1,2,3]]
 => [1,2,3] => 0
[[1,3,3]]
 => [1,2,3] => 0
[[2,2,3]]
 => [1,2,3] => 0
[[2,3,3]]
 => [1,2,3] => 0
[[3,3,3]]
 => [1,2,3] => 0
[[1,1],[3]]
 => [3,1,2] => 2
[[1,2],[3]]
 => [3,1,2] => 2
[[1,3],[2]]
 => [2,1,3] => 1
[[1,3],[3]]
 => [2,1,3] => 1
[[2,2],[3]]
 => [3,1,2] => 2
[[2,3],[3]]
 => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => 2
[[1,1,1,2]]
 => [1,2,3,4] => 0
[[1,1,2,2]]
 => [1,2,3,4] => 0
[[1,2,2,2]]
 => [1,2,3,4] => 0
[[2,2,2,2]]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [4,1,2,3] => 3
[[1,1,2],[2]]
 => [3,1,2,4] => 2
[[1,2,2],[2]]
 => [2,1,3,4] => 1
[[1,1],[2,2]]
 => [3,4,1,2] => 2
[[1,5]]
 => [1,2] => 0
[[2,5]]
 => [1,2] => 0
[[3,5]]
 => [1,2] => 0
[[4,5]]
 => [1,2] => 0
[[5,5]]
 => [1,2] => 0
[[1],[5]]
 => [2,1] => 1
[[2],[5]]
 => [2,1] => 1
[[3],[5]]
 => [2,1] => 1
[[4],[5]]
 => [2,1] => 1
Description
The number of non-left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000024
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 0
[[2,2]]
 => [1,2] => [1,0,1,0]
 => 0
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => 0
[[2,3]]
 => [1,2] => [1,0,1,0]
 => 0
[[3,3]]
 => [1,2] => [1,0,1,0]
 => 0
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => 0
[[2,4]]
 => [1,2] => [1,0,1,0]
 => 0
[[3,4]]
 => [1,2] => [1,0,1,0]
 => 0
[[4,4]]
 => [1,2] => [1,0,1,0]
 => 0
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 2
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[[1,5]]
 => [1,2] => [1,0,1,0]
 => 0
[[2,5]]
 => [1,2] => [1,0,1,0]
 => 0
[[3,5]]
 => [1,2] => [1,0,1,0]
 => 0
[[4,5]]
 => [1,2] => [1,0,1,0]
 => 0
[[5,5]]
 => [1,2] => [1,0,1,0]
 => 0
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000216
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,2] => 0
[[2,2]]
 => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [2,1] => 1
[[1,3]]
 => [1,2] => [1,2] => 0
[[2,3]]
 => [1,2] => [1,2] => 0
[[3,3]]
 => [1,2] => [1,2] => 0
[[1],[3]]
 => [2,1] => [2,1] => 1
[[2],[3]]
 => [2,1] => [2,1] => 1
[[1,1,2]]
 => [1,2,3] => [1,2,3] => 0
[[1,2,2]]
 => [1,2,3] => [1,2,3] => 0
[[2,2,2]]
 => [1,2,3] => [1,2,3] => 0
[[1,1],[2]]
 => [3,1,2] => [3,1,2] => 2
[[1,2],[2]]
 => [2,1,3] => [2,1,3] => 1
[[1,4]]
 => [1,2] => [1,2] => 0
[[2,4]]
 => [1,2] => [1,2] => 0
[[3,4]]
 => [1,2] => [1,2] => 0
[[4,4]]
 => [1,2] => [1,2] => 0
[[1],[4]]
 => [2,1] => [2,1] => 1
[[2],[4]]
 => [2,1] => [2,1] => 1
[[3],[4]]
 => [2,1] => [2,1] => 1
[[1,1,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,3,3]]
 => [1,2,3] => [1,2,3] => 0
[[2,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[2,3,3]]
 => [1,2,3] => [1,2,3] => 0
[[3,3,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,1],[3]]
 => [3,1,2] => [3,1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [3,1,2] => 2
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
 => [2,1,3] => [2,1,3] => 1
[[2,2],[3]]
 => [3,1,2] => [3,1,2] => 2
[[2,3],[3]]
 => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => 2
[[1,1,1,2]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,1,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[2,2,2,2]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [4,1,2,3] => [4,1,2,3] => 3
[[1,1,2],[2]]
 => [3,1,2,4] => [3,1,2,4] => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4,1,3,2] => 2
[[1,5]]
 => [1,2] => [1,2] => 0
[[2,5]]
 => [1,2] => [1,2] => 0
[[3,5]]
 => [1,2] => [1,2] => 0
[[4,5]]
 => [1,2] => [1,2] => 0
[[5,5]]
 => [1,2] => [1,2] => 0
[[1],[5]]
 => [2,1] => [2,1] => 1
[[2],[5]]
 => [2,1] => [2,1] => 1
[[3],[5]]
 => [2,1] => [2,1] => 1
[[4],[5]]
 => [2,1] => [2,1] => 1
Description
The absolute length of a permutation. The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Matching statistic: St000443
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
Description
The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St001007
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,2]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,3]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,4]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[1,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[2,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[3,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[4,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[5,5]]
 => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[2,2]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[2,3]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[3,3]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[2,4]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[3,4]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[4,4]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[[1,5]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[2,5]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[3,5]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[4,5]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[5,5]]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => [2,1] => 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000053
(load all 5 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 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
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 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
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 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
Description
The number of valleys of the Dyck path.
Matching statistic: St000083
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[2,2]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[1,3]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[2,3]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[3,3]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[1],[3]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[2],[3]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[1,1,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[1,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[2,2,2]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[1,1],[2]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 2
[[1,2],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[[1,4]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[2,4]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[3,4]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[4,4]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[1],[4]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[2],[4]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[3],[4]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[1,1,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[1,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[2,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[2,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[3,3,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 0
[[1,1],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 2
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[[1,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[[2,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 2
[[2,3],[3]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 2
[[1,1,1,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0
[[1,1,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0
[[1,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0
[[2,2,2,2]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 0
[[1,1,1],[2]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 3
[[1,1,2],[2]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 2
[[1,2,2],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[[1,1],[2,2]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 2
[[1,5]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[2,5]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[3,5]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[4,5]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[5,5]]
 => [1,2] => [1,0,1,0]
 => [[.,.],.]
 => 0
[[1],[5]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[2],[5]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[3],[5]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
[[4],[5]]
 => [2,1] => [1,1,0,0]
 => [.,[.,.]]
 => 1
Description
The number of left oriented leafs of a binary tree except the first one. In other other words, this is the sum of canopee vector of the tree. The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf. This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
The following 83 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000676The number of odd rises of a Dyck path. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001769The reflection length of a signed permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000379The number of Hamiltonian cycles in a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. 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. St000103The sum of the entries of a semistandard tableau. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.