searching the database
Your data matches 23 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: St000288
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1000 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 100 => 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 100 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1000 => 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 100 => 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 10 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[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]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 0 = 1 - 1
Description
The length of the partition.
Matching statistic: St000147
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [3]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [3]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [4,4,4,4,4]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 3
Description
The largest part of an integer partition.
Matching statistic: St000733
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 5
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 7
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 6
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 6
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? = 5
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 5
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[6,6,6,6],[5,5,4]]
=> [5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[7,7,7],[6,5]]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[8,8,8],[7,7]]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 7 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [4,4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[6,6,6,6],[5,5,4]]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[7,7,7],[6,5]]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[8,8,8],[7,7]]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 2 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000329
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[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 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[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 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[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]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[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]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[7,7,7],[6,6]]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[8,8],[7]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 7 - 1
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 5 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [4,4,4,3]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[6,6,6,6],[5,5,4]]
=> [5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[7,7,7],[6,5]]
=> [6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[8,8,8],[7,7]]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[9,9],[8]]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St001227
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 94%●distinct values known / distinct values provided: 62%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 94%●distinct values known / distinct values provided: 62%
Values
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[6,6,6],[5,5]]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[7,7],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[8,8],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[7,7,7],[6,6]]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[8,8],[7]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 7
[1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3,3],[2,2,2,2,2,1]]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[6,6,6,6],[5,5,4]]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[7,7,7],[6,5]]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[8,8,8],[7,7]]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[9,9],[8]]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000052
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000204
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Values
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [[[[.,.],.],.],[[.,.],.]]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [[[.,.],.],[[[.,.],.],.]]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => [[[.,.],.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => [[[.,.],.],[.,[[.,.],.]]]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1,5,6,4] => [[[.,.],.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,2,6,5,1] => [[[[.,.],.],.],[[.,.],.]]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [[.,.],[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [[.,.],[[.,.],[[.,.],.]]]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [[.,.],[[[.,.],.],[.,.]]]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [[.,.],[.,[[[.,.],.],.]]]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [[.,.],[[.,.],[[.,.],.]]]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,8,7,6,5] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,3,2,1,6,5,8,7] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [4,3,2,1,5,7,6,8] => [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,2,1,7,6,5,4,8] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,1,5,4,8,7,6] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,7,6,8] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,5,4,6,8,7] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,2,1,4,7,6,5,8] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,2,1,4,6,5,8,7] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,6,5,4,3,8] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3,8,7,6] => [[.,.],[[[.,.],.],[[[.,.],.],.]]]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,4,3,7,6,8] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,4,3,6,8,7] => [[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,3,5,8,7,6] => [[.,.],[[.,.],[.,[[[.,.],.],.]]]]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,8,7,6] => [[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4,6,8,7] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,8,7,6,5] => [[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,4,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,4,6,5,7,8] => [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,5,6,7,8,4,1] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,6,7,8,5,2,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,7,6,5,4,3,2,8] => [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,6,5,4,3,2,8,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,4,3,2,8,7,6] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,3,2,6,5,7,8] => ?
=> ? = 3
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St001167
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Values
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 3
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000356The number of occurrences of the pattern 13-2. St000065The number of entries equal to -1 in an alternating sign matrix. St001091The number of parts in an integer partition whose next smaller part has the same size. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000711The number of big exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000225Difference between largest and smallest parts in a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001330The hat guessing number of a graph.
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!