Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001697
Mapping
St001697: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => 0
[[1,2]]
 => 0
[[1],[2]]
 => 1
[[1,2,3]]
 => 0
[[1,3],[2]]
 => 2
[[1,2],[3]]
 => 1
[[1],[2],[3]]
 => 3
[[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 2
[[1,2,3],[4]]
 => 1
[[1,3],[2,4]]
 => 4
[[1,2],[3,4]]
 => 2
[[1,4],[2],[3]]
 => 4
[[1,3],[2],[4]]
 => 5
[[1,2],[3],[4]]
 => 3
[[1],[2],[3],[4]]
 => 6
[[1,2,3,4,5]]
 => 0
[[1,3,4,5],[2]]
 => 4
[[1,2,4,5],[3]]
 => 3
[[1,2,3,5],[4]]
 => 2
[[1,2,3,4],[5]]
 => 1
[[1,3,5],[2,4]]
 => 6
[[1,2,5],[3,4]]
 => 3
[[1,3,4],[2,5]]
 => 5
[[1,2,4],[3,5]]
 => 4
[[1,2,3],[4,5]]
 => 2
[[1,4,5],[2],[3]]
 => 5
[[1,3,5],[2],[4]]
 => 7
[[1,2,5],[3],[4]]
 => 4
[[1,3,4],[2],[5]]
 => 6
[[1,2,4],[3],[5]]
 => 5
[[1,2,3],[4],[5]]
 => 3
[[1,4],[2,5],[3]]
 => 6
[[1,3],[2,5],[4]]
 => 8
[[1,2],[3,5],[4]]
 => 5
[[1,3],[2,4],[5]]
 => 7
[[1,2],[3,4],[5]]
 => 4
[[1,5],[2],[3],[4]]
 => 7
[[1,4],[2],[3],[5]]
 => 8
[[1,3],[2],[4],[5]]
 => 9
[[1,2],[3],[4],[5]]
 => 6
[[1],[2],[3],[4],[5]]
 => 10
[[1,2,3,4,5,6]]
 => 0
[[1,3,4,5,6],[2]]
 => 5
[[1,2,4,5,6],[3]]
 => 4
[[1,2,3,5,6],[4]]
 => 3
[[1,2,3,4,6],[5]]
 => 2
[[1,2,3,4,5],[6]]
 => 1
[[1,3,5,6],[2,4]]
 => 8
Description
The shifted natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.
Matching statistic: St001232
Mapping
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 24%
Values
[[1]]
 => [1] => [1,0]
 => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 0
[[1],[2]]
 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 1
[[1,2,3]]
 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[[1,3],[2]]
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[1,2],[3]]
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ? = 3
[[1,2,3,4]]
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[1,3,4],[2]]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[1,2,4],[3]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[1,2,3],[4]]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2,4]]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 4
[[1,2],[3,4]]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[[1,4],[2],[3]]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 4
[[1,3],[2],[4]]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 5
[[1,2],[3],[4]]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 3
[[1],[2],[3],[4]]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 6
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,4,5],[2]]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[1,2,4,5],[3]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[1,2,3,5],[4]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[1,2,3,4],[5]]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3,5],[2,4]]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 6
[[1,2,5],[3,4]]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[[1,3,4],[2,5]]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 5
[[1,2,4],[3,5]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 4
[[1,2,3],[4,5]]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[[1,4,5],[2],[3]]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 5
[[1,3,5],[2],[4]]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,2,5],[3],[4]]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 4
[[1,3,4],[2],[5]]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 6
[[1,2,4],[3],[5]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 5
[[1,2,3],[4],[5]]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[[1,4],[2,5],[3]]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 6
[[1,3],[2,5],[4]]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 8
[[1,2],[3,5],[4]]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 5
[[1,3],[2,4],[5]]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 7
[[1,2],[3,4],[5]]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 4
[[1,5],[2],[3],[4]]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 7
[[1,4],[2],[3],[5]]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 8
[[1,3],[2],[4],[5]]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 9
[[1,2],[3],[4],[5]]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 6
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 10
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6],[2]]
 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[[1,2,4,5,6],[3]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[[1,2,3,5,6],[4]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[[1,2,3,4,6],[5]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[[1,2,3,4,5],[6]]
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,3,5,6],[2,4]]
 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 8
[[1,2,5,6],[3,4]]
 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[[1,3,4,6],[2,5]]
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 7
[[1,2,4,6],[3,5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 6
[[1,2,3,6],[4,5]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[[1,3,4,5],[2,6]]
 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 6
[[1,2,4,5],[3,6]]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 5
[[1,2,3,5],[4,6]]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 4
[[1,2,3,4],[5,6]]
 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[[1,4,5,6],[2],[3]]
 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 6
[[1,3,5,6],[2],[4]]
 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 9
[[1,2,5,6],[3],[4]]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 5
[[1,3,4,6],[2],[5]]
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 8
[[1,2,4,6],[3],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 7
[[1,2,3,6],[4],[5]]
 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 4
[[1,3,4,5],[2],[6]]
 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 7
[[1,2,4,5],[3],[6]]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 6
[[1,2,3,5],[4],[6]]
 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ? = 5
[[1,2,3,4],[5],[6]]
 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 3
[[1,3,5],[2,4,6]]
 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 9
[[1,2,5],[3,4,6]]
 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ? = 5
[[1,3,4],[2,5,6]]
 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 7
[[1,2,4],[3,5,6]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 6
[[1,2,3],[4,5,6]]
 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[[1,4,6],[2,5],[3]]
 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[1,3,6],[2,5],[4]]
 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 11
[[1,2,6],[3,5],[4]]
 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 7
[[1,3,6],[2,4],[5]]
 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 9
[[1,2,6],[3,4],[5]]
 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 5
[[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6,7],[2]]
 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[[1,2,4,5,6,7],[3]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[[1,2,3,5,6,7],[4]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[[1,2,3,4,6,7],[5]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[[1,2,3,4,5,7],[6]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[[1,2,3,4,5,6],[7]]
 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[[1,2,5,6,7],[3,4]]
 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[[1,2,3,6,7],[4,5]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[[1,2,3,4,7],[5,6]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2
[[1,2,3,7],[4,5,6]]
 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[[1,2,3,4],[5,6,7]]
 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.