Your data matches 101 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000015
(load all 6 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
Description
The number of peaks of a Dyck path.
Matching statistic: St000684
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
Description
The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$. The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$. One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$. Examples: * For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St001068
(load all 6 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 2
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 2
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 3
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
(load all 6 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000331
(load all 4 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001066
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001142
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001169
(load all 6 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001294
(load all 4 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]]. The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Matching statistic: St001296
(load all 5 compositions to match this statistic)
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001296: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => 1 = 2 - 1
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[4,3,2,1],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
Description
The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000925The number of topologically connected components of a set partition. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000783The side length of the largest staircase partition fitting into a partition. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001298The number of repeated entries in the Lehmer code of a permutation. St001432The order dimension of the partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001712The number of natural descents of a standard Young tableau. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000365The number of double ascents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.