searching the database
Your data matches 529 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001198
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
Description
The 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 $eA$.
Matching statistic: St001206
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
Description
The 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 $eA$.
Matching statistic: St000929
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1,1,1]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1,1,1]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1,1]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [6,3,3]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,4,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,2,1]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,2,2]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,2,2]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [4,3,2]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,1,1]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1]
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [4,4,1,1]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,1]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,1]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [4,2]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,3]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2,1]
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,2,1]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2,1]
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,2,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1,1]
=> ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,5,2,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,1,1,1]
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1,1,1,1]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3]
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,4,4,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,2,2]
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2]
=> ? = 2 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1]
=> 0 = 2 - 2
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000326
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [4,4]
=> 110000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [7,1,1]
=> 1000000110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,1,1]
=> 100000110 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,1,1]
=> 10000110 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> 11010 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 11110 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [6,6,1]
=> 110000010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [9,4,1]
=> 100000100010 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [8,4,1]
=> 10000100010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,2]
=> 10000001100 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [7,4,1]
=> 1000100010 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [6,2,2]
=> 100001100 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> 1000011010 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [5,5,1]
=> 11000010 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [4,2,1,1,1]
=> 100101110 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [9,5]
=> 10000100000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,3,2,2,2,1]
=> 110111010 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [8,5]
=> 1000100000 => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [9,4]
=> 10000010000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> 1000010000 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [6,6]
=> 11000000 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> 10000011110 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [7,4]
=> 100010000 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 1000011110 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [8,2,1,1]
=> 100000010110 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [3,2,2,2,2]
=> 10111100 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [9,3]
=> 10000001000 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [7,2,1,1]
=> 10000010110 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> 1011100 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [3,3,2,1,1]
=> 11010110 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [6,3]
=> 10001000 => 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 10001110 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [9,2]
=> 10000000100 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [8,2]
=> 1000000100 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [6,2,1,1]
=> 1000010110 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [3,3,1,1,1]
=> 11001110 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [5,2]
=> 1000100 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [3,2,1,1,1]
=> 10101110 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [3,3]
=> 11000 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [9,1,1,1]
=> 1000000001110 => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [8,1,1,1]
=> 100000001110 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [8,1,1]
=> 10000000110 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [7,1,1,1]
=> 10000001110 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [2,2,2,2]
=> 111100 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,1,1,1]
=> 1000001110 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,2,2,1,1]
=> 10110110 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [8]
=> 100000000 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [6]
=> 1000000 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1]
=> 111110 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ? => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [4,4,2,2,2,2,1]
=> 11001111010 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ? => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [10,5,2]
=> 1000001000100 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [9,4,4]
=> 100000110000 => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [6,3,1]
=> 100010010 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [11,7,1,1]
=> ? => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ? => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ? => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ? => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [6,2,2,2,2,2,1]
=> 1000011111010 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ? => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [8,2,2,2,1,1,1]
=> 100000011101110 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [9,6,1,1]
=> 1000100000110 => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [6,4]
=> 10010000 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [11,7,1]
=> ? => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ? => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ? => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [10,6,1]
=> 1000010000010 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> [8,8,1]
=> 11000000010 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [4,3,3,2,2,2]
=> 1011011100 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [5,2,2,2,2,2,2]
=> 100011111100 => ? = 2 - 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000296
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [4,4]
=> 110000 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [7,1,1]
=> 1000000110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,1,1]
=> 100000110 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,1,1]
=> 10000110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> 11010 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [6,6,1]
=> 110000010 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [9,4,1]
=> 100000100010 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [8,4,1]
=> 10000100010 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,2]
=> 10000001100 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [7,4,1]
=> 1000100010 => ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [6,2,2]
=> 100001100 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> 1000011010 => ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [5,5,1]
=> 11000010 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [4,2,1,1,1]
=> 100101110 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [9,5]
=> 10000100000 => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,3,2,2,2,1]
=> 110111010 => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [8,5]
=> 1000100000 => ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [9,4]
=> 10000010000 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> 1000010000 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [6,6]
=> 11000000 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> 10000011110 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [7,4]
=> 100010000 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 1000011110 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [8,2,1,1]
=> 100000010110 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [3,2,2,2,2]
=> 10111100 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [9,3]
=> 10000001000 => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [7,2,1,1]
=> 10000010110 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> 1011100 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [3,3,2,1,1]
=> 11010110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [6,3]
=> 10001000 => 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 10001110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [9,2]
=> 10000000100 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [8,2]
=> 1000000100 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [6,2,1,1]
=> 1000010110 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [3,3,1,1,1]
=> 11001110 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [5,2]
=> 1000100 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [3,2,1,1,1]
=> 10101110 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [3,3]
=> 11000 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [9,1,1,1]
=> 1000000001110 => ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [8,1,1,1]
=> 100000001110 => ? = 2 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [8,1,1]
=> 10000000110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [7,1,1,1]
=> 10000001110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [2,2,2,2]
=> 111100 => 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,1,1,1]
=> 1000001110 => 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,2,2,1,1]
=> 10110110 => 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [8]
=> 100000000 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [6]
=> 1000000 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ? => ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [4,4,2,2,2,2,1]
=> 11001111010 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [10,5,2]
=> 1000001000100 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [9,4,4]
=> 100000110000 => ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [6,3,1]
=> 100010010 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [11,7,1,1]
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [6,2,2,2,2,2,1]
=> 1000011111010 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [8,2,2,2,1,1,1]
=> 100000011101110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [9,6,1,1]
=> 1000100000110 => ? = 2 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [6,4]
=> 10010000 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [11,7,1]
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [10,6,1]
=> 1000010000010 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> [8,8,1]
=> 11000000010 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [4,3,3,2,2,2]
=> 1011011100 => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [5,2,2,2,2,2,2]
=> 100011111100 => ? = 2 - 2
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 37%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [4,4]
=> 110000 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [7,1,1]
=> 1000000110 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,1,1]
=> 100000110 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> 10000010 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,1,1]
=> 10000110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> 11010 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 110110 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 11110 => 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [6,6,1]
=> 110000010 => 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [9,4,1]
=> 100000100010 => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [8,4,1]
=> 10000100010 => 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,2,2]
=> 10000001100 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [7,4,1]
=> 1000100010 => ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [6,2,2]
=> 100001100 => 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,2,2,1]
=> 1000011010 => ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [5,5,1]
=> 11000010 => 0 = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [4,2,1,1,1]
=> 100101110 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [9,5]
=> 10000100000 => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [3,3,2,2,2,1]
=> 110111010 => 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [8,5]
=> 1000100000 => ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [9,4]
=> 10000010000 => 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> 1000010000 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [6,6]
=> 11000000 => 0 = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> 10000011110 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [7,4]
=> 100010000 => 0 = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 1000011110 => 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [8,2,1,1]
=> 100000010110 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [3,2,2,2,2]
=> 10111100 => 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [9,3]
=> 10000001000 => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [8,3]
=> 1000001000 => ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [7,2,1,1]
=> 10000010110 => 0 = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,2,2,2]
=> 1011100 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [3,3,2,1,1]
=> 11010110 => 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [6,3]
=> 10001000 => 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 10001110 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [9,2]
=> 10000000100 => 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [8,2]
=> 1000000100 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [6,2,1,1]
=> 1000010110 => 0 = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [6,2]
=> 10000100 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [3,3,1,1,1]
=> 11001110 => 0 = 2 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [5,2]
=> 1000100 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [3,2,1,1,1]
=> 10101110 => 0 = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [3,3]
=> 11000 => 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [9,1,1,1]
=> 1000000001110 => ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [8,1,1,1]
=> 100000001110 => ? = 2 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [8,1,1]
=> 10000000110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [7,1,1,1]
=> 10000001110 => 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [2,2,2,2]
=> 111100 => 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,1,1,1]
=> 1000001110 => 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,2,2,1,1]
=> 10110110 => 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [8]
=> 100000000 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [6]
=> 1000000 => 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [2,1,1]
=> 10110 => 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1]
=> 111110 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ? => ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [4,4,2,2,2,2,1]
=> 11001111010 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ? => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [10,5,2]
=> 1000001000100 => ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [9,4,4]
=> 100000110000 => ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [6,3,1]
=> 100010010 => 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [11,7,1,1]
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ? => ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [6,2,2,2,2,2,1]
=> 1000011111010 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ? => ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [8,2,2,2,1,1,1]
=> 100000011101110 => ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [9,6,1,1]
=> 1000100000110 => ? = 2 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [6,4]
=> 10010000 => 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [11,7,1]
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ? => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [10,6,1]
=> 1000010000010 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> [8,8,1]
=> 11000000010 => ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [4,3,3,2,2,2]
=> 1011011100 => ? = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [5,2,2,2,2,2,2]
=> 100011111100 => ? = 2 - 2
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St001696
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001696: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [2,2]
=> [[1,2],[3,4]]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12],[13]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> 0 = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> 0 = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [8,5]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13]]
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [8,4,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12],[13]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [7,2,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10,11],[12]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> 0 = 2 - 2
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 0 = 2 - 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> 0 = 2 - 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> 0 = 2 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ?
=> ? = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [9,4,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13],[14,15,16],[17]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ?
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [4,3,3,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13],[14,15,16],[17]]
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [8,7,1,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15],[16],[17]]
=> ? = 2 - 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ?
=> ?
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ?
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ?
=> ? = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ?
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [10,6,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ?
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [8,3,3,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13,14],[15,16],[17]]
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [10,5,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15],[16,17]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ?
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [4,3,3,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13],[14,15],[16],[17]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> [7,6,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13],[14,15],[16],[17]]
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [5,5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15],[16]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [7,4,3,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14],[15,16,17]]
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> [9,6,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17]]
=> ? = 2 - 2
Description
The natural major index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000745
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12],[13]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [8,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [8,5]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13]]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [8,4,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12],[13]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [7,2,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10,11],[12]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ?
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [9,4,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13],[14,15,16],[17]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [4,3,3,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13],[14,15,16],[17]]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [8,7,1,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15],[16],[17]]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [10,6,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [8,3,3,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13,14],[15,16],[17]]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [10,5,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15],[16,17]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [4,3,3,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13],[14,15],[16],[17]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> [7,6,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13],[14,15],[16],[17]]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1]
=> [5,5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15],[16]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> [7,4,3,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14],[15,16,17]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> [9,6,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17]]
=> ? = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> [1,1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ?
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> 1 = 2 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St001195
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 35%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,5,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,1,5,2,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,6,7,4] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,7,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,1,7,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,1,5,7] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,6,1,7,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,1,5] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6] => [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
The following 519 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001593This is the number of standard Young tableaux of the given shifted shape. St001550The number of inversions between exceedances where the greater exceedance is linked. St001139The number of occurrences of hills of size 2 in a Dyck path. St000842The breadth of a permutation. St000068The number of minimal elements in a poset. St000115The single entry in the last row. St000651The maximal size of a rise in a permutation. St000057The Shynar inversion number of a standard tableau. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000623The number of occurrences of the pattern 52341 in a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001881The number of factors of a lattice as a Cartesian product of lattices. St001625The Möbius invariant of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001130The number of two successive successions in a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000657The smallest part of an integer composition. St000659The number of rises of length at least 2 of a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001732The number of peaks visible from the left. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000232The number of crossings of a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000406The number of occurrences of the pattern 3241 in a permutation. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001394The genus of a permutation. St001513The number of nested exceedences of a permutation. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000764The number of strong records in an integer composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000636The hull number of a graph. St000862The number of parts of the shifted shape of a permutation. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001316The domatic number of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001116The game chromatic number of a graph. St001272The number of graphs with the same degree sequence. St001353The number of prime nodes in the modular decomposition of a graph. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St000268The number of strongly connected orientations of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000352The Elizalde-Pak rank of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000546The number of global descents of a permutation. St000637The length of the longest cycle in a graph. St000699The toughness times the least common multiple of 1,. St000766The number of inversions of an integer composition. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St000761The number of ascents in an integer composition. St000100The number of linear extensions of a poset. St000451The length of the longest pattern of the form k 1 2. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000079The number of alternating sign matrices for a given Dyck path. St000570The Edelman-Greene number of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001202Call 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 special CNakayama 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000455The second largest eigenvalue of a graph if it is integral. St000516The number of stretching pairs of a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001850The number of Hecke atoms of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000879The number of long braid edges in the graph of braid moves of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001261The Castelnuovo-Mumford regularity of a graph. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001306The number of induced paths on four vertices in a graph. St001356The number of vertices in prime modules of a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000069The number of maximal elements of a poset. St000124The cardinality of the preimage of the Simion-Schmidt map. St000322The skewness of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000264The girth of a graph, which is not a tree. St001260The permanent of an alternating sign matrix. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000834The number of right outer peaks of a permutation. St000878The number of ones minus the number of zeros of a binary word. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001741The largest integer such that all patterns of this size are contained in the permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000990The first ascent of a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001468The smallest fixpoint of a permutation. St000210Minimum over maximum difference of elements in cycles. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000787The number of flips required to make a perfect matching noncrossing. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001330The hat guessing number of a graph. St000617The number of global maxima of a Dyck path. St001434The number of negative sum pairs of a signed permutation. St000022The number of fixed points of a permutation. St000058The order of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000234The number of global ascents of a permutation. St001947The number of ties in a parking function. St000733The row containing the largest entry of a standard tableau. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000876The number of factors in the Catalan decomposition of a binary word. St000042The number of crossings of a perfect matching. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000007The number of saliances of the permutation. St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000392The length of the longest run of ones in a binary word. St000065The number of entries equal to -1 in an alternating sign matrix. St000297The number of leading ones in a binary word. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000920The logarithmic height of a Dyck path. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000674The number of hills of a Dyck path. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call 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 special CNakayama 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St000054The first entry of the permutation. St001725The harmonious chromatic number of a graph. St001964The interval resolution global dimension of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000741The Colin de Verdière graph invariant. St000891The number of distinct diagonal sums of a permutation matrix. St000153The number of adjacent cycles of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000729The minimal arc length of a set partition. St000925The number of topologically connected components of a set partition. St001737The number of descents of type 2 in a permutation. St000247The number of singleton blocks of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St001381The fertility of a permutation. St001536The number of cyclic misalignments of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000485The length of the longest cycle of a permutation. St000717The number of ordinal summands of a poset. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001568The smallest positive integer that does not appear twice in the partition. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001545The second Elser number of a connected graph. St000159The number of distinct parts of the integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St001432The order dimension of the partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000845The maximal number of elements covered by an element in a poset. St000627The exponent of a binary word. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000664The number of right ropes of a permutation. St000701The protection number of a binary tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000748The major index of the permutation obtained by flattening the set partition. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000035The number of left outer peaks of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001820The size of the image of the pop stack sorting operator. St000355The number of occurrences of the pattern 21-3. St001846The number of elements which do not have a complement in the lattice. St000311The number of vertices of odd degree in a graph. St000056The decomposition (or block) number of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000221The number of strong fixed points of a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001827The number of two-component spanning forests of a graph. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001871The number of triconnected components of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000091The descent variation of a composition. St000754The Grundy value for the game of removing nestings in a perfect matching. St001615The number of join prime elements of a lattice. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000662The staircase size of the code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000846The maximal number of elements covering an element of a poset. St000382The first part of an integer composition. St001665The number of pure excedances of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000359The number of occurrences of the pattern 23-1. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000401The size of the symmetry class of a permutation. St000097The order of the largest clique of the graph. St001875The number of simple modules with projective dimension at most 1. St000402Half the size of the symmetry class of a permutation. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000961The shifted major index of a permutation. St000836The number of descents of distance 2 of a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000307The number of rowmotion orbits of a poset. St000353The number of inner valleys of a permutation. St000461The rix statistic of a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000632The jump number of the poset. St001081The number of minimal length factorizations of a permutation into star transpositions. 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 $eA$. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000475The number of parts equal to 1 in a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000445The number of rises of length 1 of a Dyck path. St000648The number of 2-excedences of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!