Your data matches 220 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000306
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St001296
(load all 2 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001296: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,0,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
Description
The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
Matching statistic: St001051
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => {{1}}
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => {{1,2}}
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2 = 1 + 1
Description
The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. The bijection between set partitions of $\{1,\dots,n\}$ into $k$ blocks and trees with $n+1-k$ leaves is described in Theorem 1 of [1].
Matching statistic: St001201
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St001203
(load all 2 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St001235
(load all 3 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => [2] => [2] => 1 = 0 + 1
[1,1,0,0]
 => [2,1] => [1,1] => [1,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => [3] => [3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,3,2] => [2,1] => [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2] => [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1] => [1,1,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4] => [4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1] => [1,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,2] => [2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1] => [1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1] => [1,2,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3] => [3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,2] => [2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1] => [1,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1] => [1,2,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,2] => [2,1,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,2,1] => [1,1,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5] => [5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,3] => [3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1] => [1,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,2] => [2,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,1,1,1] => [1,2,1,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,4] => [4,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,3,1] => [1,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,2] => [2,1,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,3,1] => [1,1,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,1,1] => [1,1,2,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3] => [3,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [3,2] => [2,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,1] => [1,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,1,1] => [1,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,1,2] => [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [2,2,1] => [1,2,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,1,1,1] => [1,2,1,1] => 3 = 2 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001418
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => ? = 0
[1,0,1,0]
 => [1,2] => [1,0,1,0]
 => 0
[1,1,0,0]
 => [2,1] => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000392
(load all 7 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => => => ? = 0
[1,0,1,0]
 => [1,2] => 0 => 0 => 0
[1,1,0,0]
 => [2,1] => 1 => 1 => 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 00 => 0
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 10 => 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 01 => 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 10 => 1
[1,1,1,0,0,0]
 => [3,2,1] => 11 => 11 => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 000 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 010 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 100 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 010 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 101 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 001 => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 110 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 100 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 010 => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 011 => 101 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 011 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 101 => 110 => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 101 => 110 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 111 => 111 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0010 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0100 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0011 => 0101 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 0011 => 0101 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0110 => 1001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0101 => 1010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 0101 => 1010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 1011 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 0001 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 0110 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1100 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 0110 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => 1101 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 1000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 1010 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 0010 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0011 => 0101 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 0110 => 1001 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 0101 => 1010 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 0101 => 1010 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0111 => 1011 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => 0011 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000681
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? ∊ {0,1}
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? ∊ {0,1,1,2}
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [3,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [3,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,5] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000937
(load all 8 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0
[1,0,1,0]
 => [2,1] => [2]
 => []
 => ? ∊ {0,1}
[1,1,0,0]
 => [1,2] => [1,1]
 => [1]
 => ? ∊ {0,1}
[1,0,1,0,1,0]
 => [3,2,1] => [3]
 => []
 => ? ∊ {0,1,1,2}
[1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,0,1,0,0]
 => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,1,1,2}
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => []
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,1]
 => [1]
 => ? ∊ {0,1,1,2,2,2,3}
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5]
 => []
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => [1,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,1]
 => [1]
 => ? ∊ {0,1,2,2,2,2,2,3,3,3,4}
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,1,1]
 => [1,1]
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [3,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [4,1,1]
 => [1,1]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [3,1,1,1]
 => [1,1,1]
 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,4,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,5,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,1] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,3] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1,4] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,5] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,4,4,4,5}
Description
The number of positive values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
The following 210 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001372The length of a longest cyclic run of ones of a binary word. St001933The largest multiplicity of a part in an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000015The number of peaks of a Dyck path. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module 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. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001432The order dimension of the partition. St001638The book thickness of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001644The dimension of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001924The number of cells in an integer partition whose arm and leg length coincide. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001075The minimal size of a block of a set partition. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001233The number of indecomposable 2-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. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001118The acyclic chromatic index of a graph. St000080The rank of the poset. St000845The maximal number of elements covered by an element in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000983The length of the longest alternating subword. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St000381The largest part of an integer composition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001864The number of excedances of a signed permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000308The height of the tree associated to a permutation. St001060The distinguishing index of a graph. St000662The staircase size of the code of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. 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. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St000035The number of left outer peaks of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000120The number of left tunnels of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000328The maximum number of child nodes in a tree. St000670The reversal length of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001589The nesting number of a perfect matching. St001742The difference of the maximal and the minimal degree in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001566The length of the longest arithmetic progression in a 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 $K[x]/(x^n)$. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001624The breadth of a lattice. St000386The number of factors DDU in a Dyck path. St000871The number of very big ascents of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001435The number of missing boxes in the first row. St000891The number of distinct diagonal sums of a permutation matrix. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000703The number of deficiencies of a permutation. St000390The number of runs of ones in a binary word.