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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000015
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000678
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000982
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> 1000 => 0010 => 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1000 => 0010 => 2
([(1,2),(2,3)],4)
=> [4]
=> 10000 => 00010 => 3
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => 100010 => 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => 10010 => 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 0010 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 100100 => 100010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 10010 => 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => 00010 => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> 1000 => 0010 => 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> 100000 => 000010 => 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1000 => 0010 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1000 => 0010 => 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 100100 => 100010 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 10000 => 00010 => 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> 100100 => 100010 => 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> 10100 => 10010 => 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> 111100 => 111010 => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 110000 => 001010 => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> 10100 => 10010 => 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> 1000 => 0010 => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> 10000 => 00010 => 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> 100000 => 000010 => 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 100100 => 100010 => 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> 10000 => 00010 => 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 10000 => 00010 => 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> 110000 => 001010 => 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> 10100 => 10010 => 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> 100100 => 100010 => 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> 1000 => 0010 => 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1000 => 0010 => 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 100000 => 000010 => 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> 1000 => 0010 => 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> 111100 => 111010 => 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> 110000 => 001010 => 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> 100000 => 000010 => 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> 111100 => 111010 => 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> 111100 => 111010 => 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> 111100 => 111010 => 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> 110000 => 001010 => 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> 10100 => 10010 => 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> 10000 => 00010 => 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> 110000 => 001010 => 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> 110000 => 001010 => 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> 100100 => 100010 => 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 10100 => 10010 => 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> 110000 => 001010 => 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> 1000 => 0010 => 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> 110000 => 001010 => 2
Description
The length of the longest constant subword.
Matching statistic: St001068
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001225
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001278
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St001418
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St001499
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000052
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7)
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
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.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000331The number of upper interactions of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001910The height of the middle non-run of a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001618The cardinality of the Frattini sublattice of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001625The Möbius invariant of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000264The girth of a graph, which is not a tree. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001581The achromatic number of a graph. St001330The hat guessing number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.
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